#8096000

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=gi,f^2=i,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, 8003196);