#8505090

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

4120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412286);