#8096810

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004006);