#8463627

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

4044 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,40,38)(35,45,46,41)(36,43,37,42)(39,44,48,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,41)(36,44)(37,47)(38,45)(39,43)(40,46)(42,48);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,40,37)(34,42,38,43)(35,44,46,47)(39,41,48,45);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,40,36)(34,43,38,42)(35,44,46,47)(39,41,48,45);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,40,34)(35,45,46,41)(36,43,37,42)(39,47,48,44);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,39)(34,44)(35,43)(36,41)(37,45)(38,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=g,e^2=hi,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, 8370823);