#8464034

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

4149 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)(8,11)(10,15)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(34,40)(35,44)(36,43)(42,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(18,23)(19,26)(24,30)(27,32)(33,34)(35,41)(36,46)(37,42)(38,43)(39,40)(44,47)(45,48);(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,35,39,44)(34,41,40,47)(36,45,43,37)(38,42,46,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,36)(34,38)(35,45)(37,44)(39,43)(40,46)(41,42)(47,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,37,39,45)(34,42,40,48)(35,36,44,43)(38,41,46,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,38)(34,43)(35,42)(36,40)(37,41)(39,46)(44,48)(45,47)

Polycyclic presentation:

2^9|c^2=hi,e^2=gi,f^2=i,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, 8371230);