#8467143

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=h,e^2=ghi,f^2=i,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, 8374339);