#9478147

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9385343);