#9243499

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	#53770	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3849 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,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, 9150695);