#10247561

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

4172 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,39,44)(35,42,45,40)(36,41,37,46)(38,43,48,47);(2,3)(4,11)(8,10)(12,13)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(33,35)(34,40)(36,43)(37,47)(38,41)(39,45)(42,44)(46,48);(2,3)(7,9)(8,10)(15,16)(18,25)(19,28)(20,30)(26,32)(33,36,39,37)(34,41,44,46)(35,43,45,47)(38,42,48,40);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,37,39,36)(34,41,44,46)(35,43,45,47)(38,40,48,42);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,35)(34,42)(36,43)(37,47)(38,46)(39,45)(40,44)(41,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(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=gh,d^2=gi,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, 10154757);