#6844199

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(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);(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,32)(22,29)(24,38)(25,41)(27,40)(31,44)(34,35)(36,45)(39,42)(43,46);(2,6)(7,8)(11,16)(13,15)(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);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,23)(20,34)(24,27)(26,37)(30,33)(32,35)(38,40)(43,46);(1,4,5,3)(2,8,6,7)(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=fi,c^2=fh,d^2=gi,e^2=fgh,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, 6751395);