#9002489

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,e^2=ghi,f^2=gh,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, 8909685);