#7107810

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

1240 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=hi,d^2=i,e^2=fi,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=dhi,e^c=ei,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, 7015006);