#8445949

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

4022 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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, 8353145);