#6894208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37418	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6801404);