#10349009

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

4184 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)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,18)(19,28)(20,24)(21,25)(22,29)(23,27)(26,32)(30,31)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(2,6)(3,10)(8,14)(11,13)(19,20)(22,32)(24,28)(26,29)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(4,12)(8,14)(11,13)(15,16)(17,19,23,20)(18,24,27,28)(21,26,31,29)(22,30,32,25)(33,36,38,44)(34,39,41,35)(37,43,46,47)(40,45,48,42);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,20,23,19)(18,24,27,28)(21,26,31,29)(22,25,32,30)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,4)(2,8)(3,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,21,23,31)(18,25,27,30)(19,29,20,26)(22,24,32,28)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,22,23,32)(18,26,27,29)(19,30,20,25)(21,24,31,28)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=h,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, 10256205);