#6788734

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^2=hi,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, 6695930);