#7107874

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1235 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)(27,36)(28,33)(29,34)(30,32)(31,37)(35,43)(38,48)(39,44)(40,45)(41,49)(42,47)(46,51)(50,54)(52,55)(53,56);(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)(26,31)(28,40)(29,37)(30,39)(32,46)(33,48)(34,36)(35,44)(38,41)(42,53)(43,51)(45,49)(47,55)(50,56)(52,54);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(27,37)(30,43)(31,44)(33,47)(34,35)(36,46)(38,40)(42,54)(45,55)(48,56)(49,50);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,41)(26,33,34,49)(27,38,37,40)(30,42,43,54)(31,45,36,48)(32,47,35,50)(39,52,51,53)(44,55,46,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(26,34)(28,41)(31,36)(32,35)(38,40)(42,54)(44,46)(52,53)

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=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7015070);