#471019

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16732
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

537 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)(3,8,15,28)(4,17,20,44)(5,21,23,10)(6,12)(7,11)(9,19,30,47)(13,34)(14,38)(16,27,42,54)(18,43)(22,48,49,32)(24,33)(25,50)(26,40)(29,55)(31,45)(35,58,39,62)(36,59)(37,52)(41,61)(46,57)(51,60,56,63)(53,64);(1,3)(2,8,11,28)(4,13,20,36)(5,14)(6,16)(7,15)(9,25)(10,26,21,38)(12,27,33,54)(17,34)(18,35,46,39)(19,37,47,50)(22,41)(23,40)(24,42)(29,51)(30,52)(31,56)(32,53,48,61)(43,58)(44,59)(45,60,55,63)(49,64)(57,62);(1,4)(2,9,11,30)(3,13,15,36)(5,18)(6,19)(7,20)(8,25)(10,31,21,29)(12,17,33,44)(14,39,40,35)(16,37,42,50)(22,45)(23,46)(24,47)(26,51)(27,34)(28,52)(32,57,48,43)(38,56)(41,63,64,60)(49,55)(53,58)(54,59)(61,62);(1,5,6,22,7,23,24,49)(2,10,12,32,11,21,33,48)(3,14,16,41,15,40,42,64)(4,18,19,45,20,46,47,55)(8,26,27,53,28,38,54,61)(9,29,17,43,30,31,44,57)(13,35,37,60,36,39,50,63)(25,51,34,58,52,56,59,62)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378215);