#8455999

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

4202 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)(3,8,11,7)(4,5,9,15)(12,14,16,13)(18,23)(19,28)(26,30)(27,32)(34,37)(36,40)(41,45)(44,46);(3,11)(7,8)(12,16)(13,14)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19)(18,24)(20,27)(21,26)(22,28)(23,25)(29,32)(30,31)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,19)(18,25)(20,27)(21,30)(22,28)(23,24)(26,31)(29,32)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,4,6,9)(2,5,10,15)(3,12,11,16)(7,13,8,14)(17,20,22,29)(18,26,23,30)(19,27,28,32)(21,25,31,24)(33,37,39,34)(35,36,43,40)(38,45,47,41)(42,44,48,46);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,21,22,31)(18,27,23,32)(19,26,28,30)(20,25,29,24)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8363195);