#9490114

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

3985 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=gh,e^2=h,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,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, 9397310);