#10247705

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

4182 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154901);