#10340233

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

4036 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18,22,25)(19,23,27,20)(21,24,30,32)(26,31,29,28)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,2,6,4)(3,8,11,7)(5,9,13,12)(10,15,16,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,28)(26,30)(31,32)(33,34,38,37)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19,22,27)(18,23,25,20)(21,26,30,29)(24,31,32,28)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,37,38,34)(35,41,43,39)(36,40,44,45)(42,46,48,47);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47)

Polycyclic presentation:

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

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, 10247429);