#8462100

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

4099 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(20,28)(25,29)(26,32)(30,31)(33,34,40,38)(35,45,47,41)(36,43,37,42)(39,44,46,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(17,18,22,21)(19,24,27,23)(20,25,28,29)(26,31,32,30)(33,35)(34,41)(36,46)(37,39)(38,45)(40,47)(42,48)(43,44);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,19)(18,23)(20,26)(21,24)(22,27)(25,30)(28,32)(29,31)(33,36,40,37)(34,42,38,43)(35,46,47,39)(41,48,45,44);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,37,40,36)(34,43,38,42)(35,46,47,39)(41,48,45,44);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,20,22,28)(18,25,21,29)(19,26,27,32)(23,30,24,31)(33,38,40,34)(35,45,47,41)(36,43,37,42)(39,48,46,44);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,21,22,18)(19,24,27,23)(20,29,28,25)(26,31,32,30)(33,39,40,46)(34,44,38,48)(35,37,47,36)(41,43,45,42)

Polycyclic presentation:

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

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, 8369296);