#8696262

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

4186 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)(3,10)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(18,20)(19,25)(24,28)(27,29)(33,34,39,43)(35,40,46,47)(36,45,37,41)(38,44,48,42);(1,3,6,5)(2,7,9,10)(4,13,14,11)(8,16,12,15)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,35)(34,40)(36,42)(37,44)(38,45)(39,46)(41,48)(43,47);(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36,39,37)(34,41,43,45)(35,44,46,42)(38,47,48,40);(1,4,6,14)(2,8,9,12)(3,11,5,13)(7,15,10,16)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,37,39,36)(34,41,43,45)(35,44,46,42)(38,40,48,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,20,22,18)(19,26,25,23)(21,24,30,28)(27,31,29,32)(33,38,39,48)(34,42,43,44)(35,45,46,41)(36,47,37,40);(1,5,6,3)(2,7,9,10)(4,11,14,13)(8,16,12,15)(17,21,22,30)(18,24,20,28)(19,27,25,29)(23,31,26,32)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

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, 8603458);