#10464525

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

4223 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,e^2=h,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, 10371721);