#8425514

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

4193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332710);