#9267005

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

3989 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)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(17,19,23,20)(18,24,21,25)(22,28,30,27)(26,32,29,31)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(2,9)(3,12)(4,14)(10,16)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,46)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9174201);