#6788771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

1200 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26)(27,35)(28,39)(29,31)(30,44)(32,38)(33,41)(34,43)(36,50)(37,52)(40,49)(42,48)(45,51)(46,56)(47,54)(53,55);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(26,31)(27,36)(33,39)(34,44)(37,40)(38,51)(48,55)(54,56);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,27,30,38,29,36,43,51)(26,32,34,35,31,45,44,50)(28,40,42,56,41,37,53,54)(33,47,48,52,39,46,55,49);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,39)(32,46)(34,48)(35,49)(36,40)(38,54)(43,53)(44,55)(45,47)(50,52)(51,56);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(28,41)(33,39)(37,40)(42,53)(46,47)(48,55)(49,52)(54,56)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6695967);