#6752338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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

1214 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6659534);