#8810136

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717332);