#9240918

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

4155 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=gh,f^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, 9148114);