#8976979

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

4067 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8884175);