#8445939

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

4009 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=i,f^2=g,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, 8353135);