#475246

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17038
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:

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

Polycyclic presentation:

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

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, 382442);