#7180316

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1208 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)(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,37)(35,43)(38,48)(39,44)(40,45)(41,50)(42,47)(46,51)(49,54)(52,55)(53,56);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,14)(13,20)(16,24)(17,18)(19,22)(21,23)(25,27,29,37)(26,31,34,36)(28,40,41,38)(30,39,43,51)(32,46,35,44)(33,48,50,45)(42,53,54,52)(47,55,49,56);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(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)(6,7)(11,12)(15,18)(19,20)(23,24)(25,28,29,41)(26,33,34,50)(27,38,37,40)(30,42,43,54)(31,45,36,48)(32,47,35,49)(39,52,51,53)(44,55,46,56);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(26,34)(27,37)(28,41)(32,35)(39,51)(42,54)(45,48)(55,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=fhi,c^2=fi,d^2=i,e^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, 7087512);