#8733728

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8640924);