#6760623

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

1204 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)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,30,35,29,34,43,32)(27,36,39,45,38,31,44,47)(28,40,42,48,41,33,49,50)(37,46,53,56,52,51,54,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,29,38)(26,31,34,36)(28,37,41,52)(30,39,43,44)(32,45,35,47)(33,46,40,51)(42,53,49,54)(48,55,50,56);(1,3,4,8)(2,5,6,7)(26,32)(30,43)(31,45)(33,48)(34,35)(36,47)(39,44)(40,50)(42,49)(46,55)(51,56)(53,54);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,46)(32,48)(34,40)(35,50)(36,51)(38,52)(39,53)(43,49)(44,54)(45,55)(47,56);(2,6)(5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,38)(28,41)(31,36)(33,40)(39,44)(42,49)(45,47)(48,50)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6667819);