#8464208

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

4153 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,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, 8371404);