#9243558

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
prime radical	#54183	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,f^2=g,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, 9150754);