#6652678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1197 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6559874);