#6487238

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394434);