#10424755

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

4080 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10331951);