#6776887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1253 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6684083);