#451758

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16012
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358954);