#8455918

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

4164 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)(17,18,23,27)(19,24,30,28)(20,22,26,32)(21,31,29,25)(33,34,38,42)(35,39,37,41)(36,44,45,40)(43,48,46,47);(1,3,6,9)(2,5,10,7)(4,11,13,15)(8,12,16,14)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,23,30)(18,24,27,28)(20,29,26,21)(22,25,32,31)(33,35,38,37)(34,39,42,41)(36,43,45,46)(40,47,44,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,20)(18,22)(19,29)(21,30)(23,26)(24,25)(27,32)(28,31)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,48)(42,44);(1,4,6,13)(2,8,10,16)(3,11,9,15)(5,12,7,14)(17,21,23,29)(18,25,27,31)(19,20,30,26)(22,24,32,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,5)(2,9)(3,10)(4,12)(6,7)(8,15)(11,16)(13,14)(17,22)(18,26)(19,25)(20,27)(21,24)(23,32)(28,29)(30,31)(33,37,38,35)(34,41,42,39)(36,43,45,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,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, 8363114);