#8463499

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

4015 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(35,43)(36,45)(39,46)(40,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(19,29)(20,31)(24,28)(25,30)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35,38,43)(34,39,42,46)(36,44,45,37)(40,47,48,41);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25)(33,36)(34,40)(35,44)(37,43)(38,45)(39,47)(41,46)(42,48);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37,38,44)(34,41,42,47)(35,36,43,45)(39,40,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=g,e^2=i,f^2=hi,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, 8370695);