#509608

order	512
nilpotency class	4
isoclinism	isoclinic class of #509608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30846
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,12)(4,20)(5,10)(6,31)(7,15)(8,14)(9,16)(11,24)(13,27)(17,49)(18,44)(19,45)(21,40)(22,55)(23,59)(25,42)(26,48)(28,43)(29,37)(30,38)(32,47)(33,46)(34,36)(35,54)(39,58)(41,61)(50,63)(51,52)(53,56)(57,62)(60,64);(1,3)(2,10)(4,21)(5,27)(6,29)(7,18)(8,28)(9,19)(11,39)(12,31)(13,43)(14,36)(15,37)(16,38)(17,25)(20,35)(22,23)(24,51)(26,50)(30,32)(33,34)(40,59)(41,63)(42,55)(44,47)(45,46)(48,62)(49,64)(52,53)(54,56)(57,60)(58,61);(1,4)(2,11)(3,17)(5,22)(6,23)(7,25)(8,24)(9,26)(10,35)(12,40)(13,41)(14,48)(15,20)(16,42)(18,50)(19,51)(21,28)(27,56)(29,57)(30,58)(31,53)(32,60)(33,61)(34,54)(36,39)(37,55)(38,62)(43,52)(44,49)(45,63)(46,64)(47,59);(1,5,27,3,8,29,33,18,9,30,32,19,7,34,6,28)(2,12,31,10,15,43,47,36,16,45,46,38,14,44,13,37)(4,22,56,17,24,57,61,50,26,58,60,51,25,54,23,21)(11,40,53,35,20,52,59,39,42,63,64,62,48,49,41,55)

Polycyclic presentation:

2^9|d^2=eh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416804);