#7125761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7032957);