#9863412

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770608);