#8461728

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

4195 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=i,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, 8368924);