#8455936

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

4179 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)(3,5,9,7)(4,8,13,16)(11,12,15,14)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28)(33,34,39,42)(35,40,37,36)(38,41,47,46)(43,48,45,44);(1,3,6,9)(2,5,10,7)(4,11,13,15)(8,12,16,14)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31)(33,35,39,37)(34,36,42,40)(38,43,47,45)(41,44,46,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,5)(4,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,48)(44,45);(1,4,6,13)(2,8,10,16)(3,11,9,15)(5,12,7,14)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,48,46,44);(1,5)(2,9)(3,10)(4,12)(6,7)(8,15)(11,16)(13,14)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)(33,38,39,47)(34,41,42,46)(35,43,37,45)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,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, 8363132);