#6909180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907869
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816376);