#8906999

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

4301 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)(4,7,12,9)(5,10,13,8)(11,15,16,14)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(33,34,38,39)(35,41,44,40)(36,37,45,48)(42,43,46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(34,39)(36,45)(40,41)(42,46);(2,3)(5,13)(7,9)(11,16)(19,27)(20,29)(23,30)(24,32)(33,35,38,44)(34,40,39,41)(36,42,45,46)(37,47,48,43);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,35)(34,40)(36,46)(37,43)(38,44)(39,41)(42,45)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,36)(34,37)(35,42)(38,45)(39,48)(40,47)(41,43)(44,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,37)(34,36)(35,43)(38,48)(39,45)(40,46)(41,42)(44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8814195);