#8461700

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

4166 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)(3,7)(4,9)(5,10)(6,11)(8,13)(12,14)(15,16)(20,28)(25,29)(26,32)(30,31)(33,34,39,43)(35,44,47,40)(36,38,42,48)(37,46,45,41);(1,3,6,13)(2,7,11,8)(4,10,14,16)(5,12,15,9)(17,18,22,21)(19,24,27,23)(20,25,28,29)(26,31,32,30)(33,35,39,47)(34,40,43,44)(36,45,42,37)(38,46,48,41);(1,3)(2,8)(4,10)(5,12)(6,13)(7,11)(9,15)(14,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,30)(28,32)(29,31)(33,36,39,42)(34,38,43,48)(35,45,47,37)(40,46,44,41);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,15)(14,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,34)(35,40)(36,38)(37,41)(39,43)(42,48)(44,47)(45,46);(1,4,6,14)(2,9,11,12)(3,10,13,16)(5,8,15,7)(17,20,22,28)(18,25,21,29)(19,26,27,32)(23,30,24,31)(33,37)(34,41)(35,36)(38,40)(39,45)(42,47)(43,46)(44,48);(1,5)(2,10)(3,12)(4,8)(6,15)(7,14)(9,13)(11,16)(17,21,22,18)(19,24,27,23)(20,29,28,25)(26,31,32,30)(33,38)(34,42)(35,46)(36,43)(37,40)(39,48)(41,47)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=i,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, 8368896);