#6880094

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,35)(28,39)(29,33)(30,44)(31,37)(32,42)(34,43)(36,50)(38,46)(40,54)(41,53)(45,55)(47,51)(48,56)(49,52);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,31)(28,40)(29,37)(30,38)(32,47)(33,35)(34,46)(36,41)(39,50)(42,55)(43,52)(44,49)(45,48)(51,56)(53,54);(2,6)(7,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,30,42,29,41,43,56)(26,32,34,39,33,48,44,53)(27,36,38,51,37,40,52,55)(31,45,46,54,35,47,49,50);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,33)(27,37)(28,41)(34,44)(38,52)(42,56)(45,47)(50,54);(1,4,5,3)(2,7,6,8)(11,21)(15,18)(19,20)(23,24)(28,41)(32,48)(36,40)(39,53)(42,56)(45,47)(50,54)(51,55)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=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, 6787290);