#8455934

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,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, 8363130);