#6788727

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	#31224	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(26,30)(27,33)(28,36)(31,35)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,27,26,31,29,35,30,33)(28,34,32,38,36,40,37,39);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(28,36)(32,37)(34,40)(38,39)

Polycyclic presentation:

2^9|b^2=f,c^2=fg,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313330, #72313593, #72313904
maximal normal subgroup of	#72313330, #72313593, #72313904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6695923);