#489354

order	512
nilpotency class	4
isoclinism	isoclinic class of #489351
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18524
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61111	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

886 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgi,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,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306810, #72306811, #72306815
maximal normal subgroup of	#72306810, #72306811, #72306815

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396550);