#8762112

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

4276 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,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, 8669308);