#8908439

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

4050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815635);