#10325232

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

4248 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)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28)(34,39)(35,41)(37,45)(44,47);(3,4)(5,14)(7,10)(8,13)(33,34,38,39)(35,36,41,40)(37,42,45,46)(43,44,48,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28)(33,35)(34,36)(37,43)(38,41)(39,40)(42,44)(45,48)(46,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30)(33,36)(34,41)(35,39)(37,44)(38,40)(42,48)(43,46)(45,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gh,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, 10232428);