#8527990

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8435186);