#6832584

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

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fh,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6739780);