#6818204

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1453 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,33)(21,27)(22,26)(23,28)(24,31)(25,32)(30,45)(34,40)(35,43)(36,41)(37,42)(38,44)(39,46)(47,48);(2,5)(3,7)(17,19,22,31)(18,24,26,29)(20,34,32,23)(21,35,37,36)(25,40,33,28)(27,41,42,43)(30,38,46,47)(39,44,45,48);(9,11,12,16)(10,13,14,15)(17,20,23,19,22,32,34,31)(18,25,28,24,26,33,40,29)(21,36,38,46,37,35,47,30)(27,43,44,45,42,41,48,39);(2,5)(6,8)(18,26)(21,37)(24,29)(25,33)(28,40)(30,46)(35,36)(38,47);(1,3,4,7)(2,6,5,8)(17,21,34,47,22,37,23,38)(18,27,40,48,26,42,28,44)(19,30,20,35,31,46,32,36)(24,39,25,41,29,45,33,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,e^2=fh,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, 6725400);