#7107953

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

1237 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)(25,26,30,35)(27,37,38,32)(28,34,44,52)(29,36,31,33)(39,50,42,47)(40,48,41,46)(43,49,45,51)(53,55,54,56);(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)(26,32)(28,42)(29,40)(30,38)(31,41)(33,48)(34,50)(35,37)(36,46)(39,44)(43,54)(45,53)(47,52)(49,55)(51,56);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,33)(27,38)(29,31)(32,46)(34,49)(35,36)(37,48)(39,42)(43,45)(47,55)(50,56)(51,52);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,44)(26,34,35,52)(27,39,38,42)(29,45,31,43)(32,47,37,50)(33,49,36,51)(40,54,41,53)(46,55,48,56);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,33,35,36)(27,40,38,41)(28,43,44,45)(32,48,37,46)(34,51,52,49)(39,53,42,54)(47,55,50,56)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=hi,d^2=i,e^2=f,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,e^c=ei,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, 7015149);