#10534351

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

4304 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)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(34,39)(35,40)(37,45)(43,46);(1,3)(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(34,39)(35,40)(41,47)(42,48);(1,4,8,5)(2,9,6,3)(7,12,15,13)(10,16,14,11)(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,34,38,39)(35,36,40,44)(37,41,45,47)(42,43,48,46);(1,5,8,4)(2,9,6,3)(7,13,15,12)(10,16,14,11)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,48)(46,47);(1,6,8,2)(3,5,9,4)(7,10,15,14)(11,12,16,13)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,45)(47,48);(1,7)(2,10)(3,11)(4,12)(5,13)(6,14)(8,15)(9,16)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,37)(34,41)(35,42)(36,43)(38,45)(39,47)(40,48)(44,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=ghi,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, 10441547);