#454976

order	512
nilpotency class	3
isoclinism	isoclinic class of #454975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16258
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285398, #72286154, #72286176
maximal normal subgroup of	#72285398, #72286154, #72286176

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362172);