#6867535

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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)(3,7)(4,6)(5,8)(17,18,22,26,23,27,21,35)(19,24,30,41,29,39,36,40)(20,31,33,44,34,43,37,25)(28,45,47,42,32,48,46,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,32)(21,36)(22,30)(23,29)(25,42)(26,41)(27,39)(28,34)(31,48)(33,46)(35,40)(37,47)(38,44)(43,45);(10,14)(15,16)(17,20,23,34)(18,25,27,44)(19,28,29,32)(21,33,22,37)(24,38,39,42)(26,43,35,31)(30,46,36,47)(40,45,41,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,29)(24,39)(28,32)(30,36)(38,42)(40,41)(45,48)(46,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,27,35)(19,30,29,36)(20,33,34,37)(24,40,39,41)(25,43,44,31)(28,46,32,47)(38,45,42,48)

Polycyclic presentation:

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

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, 6774731);