#8718209

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

4213 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625405);