#8691013

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

4197 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598209);