#8471108

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

4127 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)(17,18,23,20)(19,24,30,27)(21,31,29,25)(22,26,28,32)(34,37)(36,40)(41,45)(44,46);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,18,23,20)(19,27,30,24)(21,25,29,31)(22,26,28,32)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,20,23,18)(19,27,30,24)(21,31,29,25)(22,26,28,32)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,6,8,3)(2,7,11,9)(4,13,5,12)(10,15,14,16)(17,21,23,29)(18,25,20,31)(19,28,30,22)(24,26,27,32)(33,37,39,34)(35,36,43,40)(38,45,47,41)(42,44,48,46);(1,7)(2,3)(4,15)(5,16)(6,11)(8,9)(10,12)(13,14)(17,22,23,28)(18,26,20,32)(19,29,30,21)(24,25,27,31)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=i,e^2=gi,f^2=ghi,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, 8378304);