#488873

order	512
nilpotency class	4
isoclinism	isoclinic class of #488870
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1619
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61105	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

951 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 951 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,14)(3,9,18,34)(4,19,23,53)(5,13,7,11)(6,25,29,39)(10,35,38,22)(12,24,42,28)(15,44,47,64)(16,43,30,32)(17,40,50,62)(20,52,21,41)(26,33,55,60)(27,37,54,36)(31,56,58,46)(45,51,48,63)(49,57,61,59);(1,3)(2,9)(4,15)(5,16)(6,26)(7,30)(8,18)(10,31)(11,32)(12,40)(13,43)(14,34)(17,28)(19,51)(20,45)(21,48)(22,49)(23,47)(24,50)(25,33)(27,46)(29,55)(35,61)(36,57)(37,59)(38,58)(39,60)(41,44)(42,62)(52,64)(53,63)(54,56);(1,4)(2,10)(3,15)(5,20)(6,27)(7,21)(8,23)(9,31)(11,36)(12,41)(13,37)(14,38)(16,45)(17,49)(18,47)(19,25)(22,28)(24,35)(26,46)(29,54)(30,48)(32,57)(33,51)(34,58)(39,53)(40,44)(42,52)(43,59)(50,61)(55,56)(60,63)(62,64);(1,5,8,7)(2,11,14,13)(3,16,18,30)(4,20,23,21)(6,24,29,28)(9,32,34,43)(10,36,38,37)(12,39,42,25)(15,45,47,48)(17,26,50,55)(19,41,53,52)(22,27,35,54)(31,57,58,59)(33,40,60,62)(44,63,64,51)(46,61,56,49)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306815, #72306834
maximal normal subgroup of	#72306815, #72306834

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396069);