#8755154

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

3971 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662350);