#6820330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727526);