#9240974

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

4005 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,39)(35,41,43,37)(36,44,45,40)(42,46,48,47);(2,7)(5,8)(9,14)(13,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,43)(34,37,39,41)(36,42,45,48)(40,46,44,47);(4,12)(9,14)(10,16)(13,15)(18,19)(23,25)(24,26)(30,31)(34,39)(36,45)(37,41)(42,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,45)(34,40,39,44)(35,42,43,48)(37,46,41,47);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37)(34,35)(36,46)(38,41)(39,43)(40,42)(44,48)(45,47);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,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, 9148170);