#7107881

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,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,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, 7015077);