#8463388

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(17,18,24,27)(19,26,22,25)(20,23,21,31)(28,32,29,30)(33,34,40,38)(35,45,47,41)(36,43,37,42)(39,44,46,48);(3,11)(7,8)(12,16)(13,14)(17,19,24,22)(18,25,27,26)(20,28,21,29)(23,30,31,32)(33,35)(34,41)(36,46)(37,39)(38,45)(40,47)(42,48)(43,44);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,20,24,21)(18,23,27,31)(19,28,22,29)(25,30,26,32)(33,36,40,37)(34,42,38,43)(35,46,47,39)(41,48,45,44);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,21,24,20)(18,23,27,31)(19,29,22,28)(25,30,26,32)(33,37,40,36)(34,43,38,42)(35,46,47,39)(41,48,45,44);(1,4,6,9)(2,5,10,15)(3,12,11,16)(7,13,8,14)(17,22,24,19)(18,26,27,25)(20,29,21,28)(23,32,31,30)(33,38,40,34)(35,45,47,41)(36,43,37,42)(39,48,46,44);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,23)(18,21)(19,30)(20,27)(22,32)(24,31)(25,29)(26,28)(33,39,40,46)(34,44,38,48)(35,37,47,36)(41,43,45,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=hi,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, 8370584);