#8445951

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
prime radical	#41081	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=gi,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, 8353147);