#10480831

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388027);