#6777261

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1243 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6684457);