#8096885

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

4221 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)(17,18,22,23)(19,20,25,24)(21,26,30,31)(27,28,32,29)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(18,23)(19,25)(26,31)(27,32)(36,44)(40,45)(42,48)(46,47);(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(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)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(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)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,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, 8004081);