#6820380

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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)(19,27)(20,25)(21,23)(22,26)(24,31)(28,38)(29,37)(30,42)(32,45)(33,39)(34,36)(35,41)(40,44)(43,47)(46,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(19,28)(20,32)(22,29)(24,38)(25,41)(27,40)(31,44)(34,35)(36,45)(39,42)(43,46);(9,12,13,11)(10,14,15,16)(17,19,29,44,21,31,22,28)(18,24,37,38,23,27,26,40)(20,33,32,43,34,30,35,46)(25,42,45,48,36,39,41,47);(1,3,4,7)(2,6,5,8)(18,23)(20,34)(24,27)(26,37)(30,33)(32,35)(38,40)(43,46);(2,5)(6,8)(17,20,22,35,21,34,29,32)(18,25,26,41,23,36,37,45)(19,30,28,43,31,33,44,46)(24,39,40,48,27,42,38,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6727576);