#8932749

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

4140 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839945);