#7040304

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1190 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)(25,26,31,37)(27,32,41,49)(28,34,43,52)(29,36,30,35)(33,45,51,42)(38,54,39,47)(40,53,44,48)(46,55,50,56);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,41)(26,32,37,49)(28,39,43,38)(29,44,30,40)(33,50,51,46)(34,47,52,54)(35,53,36,48)(42,55,45,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,43)(26,33,37,51)(27,38,41,39)(29,42,30,45)(32,46,49,50)(34,36,52,35)(40,55,44,56)(47,53,54,48);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,39)(29,45)(30,42)(31,43)(32,47)(33,35)(36,51)(37,52)(38,41)(40,55)(44,56)(46,48)(49,54)(50,53);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,35,37,36)(27,40,41,44)(28,42,43,45)(32,48,49,53)(33,34,51,52)(38,55,39,56)(46,47,50,54)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=f,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=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, 6947500);