#8096780

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003976);