#9508567

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

3990 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,43)(35,39,46,48)(36,41,45,40)(37,42,44,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,46)(34,39,43,48)(36,44,45,37)(40,42,41,47);(2,6)(3,10)(4,13)(14,15)(18,19)(23,25)(24,26)(30,31)(33,36,38,45)(34,40,43,41)(35,37,46,44)(39,47,48,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,43)(42,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37,38,44)(34,42,43,47)(35,45,46,36)(39,40,48,41);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9415763);