#9668375

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

4396 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575571);