#6832552

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 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)(21,26)(22,27)(23,28)(25,31)(29,38)(30,39)(33,43)(34,41)(35,40)(36,42)(37,44)(45,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,39)(20,33,31,23)(21,35,36,34)(25,28,32,43)(26,40,42,41)(29,45,46,37)(38,47,48,44);(2,5)(6,8)(10,14)(11,16)(17,20,23,19,22,31,33,30)(18,25,28,39,27,32,43,24)(21,35,37,46,36,34,45,29)(26,41,44,38,42,40,47,48);(10,14)(13,15)(21,36)(26,42)(29,46)(34,35)(37,45)(38,48)(40,41)(44,47);(1,3,4,7)(2,6,5,8)(17,21,33,45,22,36,23,37)(18,26,43,47,27,42,28,44)(19,29,20,34,30,46,31,35)(24,38,32,41,39,48,25,40)

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,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, 6739748);