#8505641

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

4032 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)(3,11)(4,5)(6,8)(7,10)(9,15)(12,14)(13,16)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,38)(35,40,46,45)(36,42,37,41)(43,48,44,47);(1,3,7,6)(2,8,10,11)(4,12,15,13)(5,16,9,14)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,39,46)(34,40,38,45)(36,43,37,44)(41,47,42,48);(1,4)(2,9)(3,12)(5,10)(6,13)(7,15)(8,14)(11,16)(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,36,39,37)(34,41,38,42)(35,43,46,44)(40,47,45,48);(1,4)(2,9)(3,13)(5,10)(6,12)(7,15)(8,16)(11,14)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,5,7,9)(2,4,10,15)(3,14,6,16)(8,13,11,12)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37,39,36)(34,42,38,41)(35,44,46,43)(40,48,45,47);(1,6,7,3)(2,8,10,11)(4,13,15,12)(5,16,9,14)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38,39,34)(35,45,46,40)(36,42,37,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=i,e^2=h,f^2=hi,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, 8412837);