#8772701

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

4240 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,16)(14,15)(17,18,22,26)(19,24,29,23)(20,30,25,21)(27,32,31,28)(34,37)(39,41)(40,45)(46,47);(2,7)(5,8)(9,15)(13,16)(17,18)(19,23)(20,21)(22,26)(24,29)(25,30)(27,32)(28,31)(34,37)(36,44)(39,41)(42,48);(1,3,6,11)(2,5,7,8)(4,12,14,10)(9,16,15,13)(17,19,22,29)(18,23,26,24)(20,31,25,27)(21,28,30,32)(33,34,38,37)(35,41,43,39)(36,45,44,40)(42,46,48,47);(1,3)(2,8)(4,10)(5,7)(6,11)(9,16)(12,14)(13,15)(17,20,22,25)(18,21,26,30)(19,27,29,31)(23,32,24,28)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,4,6,14)(2,9,7,15)(3,10,11,12)(5,13,8,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32)(33,36,38,44)(34,40,37,45)(35,42,43,48)(39,46,41,47);(1,5,6,8)(2,3,7,11)(4,13,14,16)(9,10,15,12)(17,21)(18,25)(19,28)(20,26)(22,30)(23,27)(24,31)(29,32)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=h,e^2=gh,f^2=gh,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, 8679897);