#9243508

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

3999 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)(3,10)(4,7)(5,8)(6,12)(9,15)(11,13)(14,16)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34)(35,44)(36,42)(37,39)(38,41)(40,43)(45,46)(47,48);(1,3,7,8)(2,5,4,10)(6,14,15,11)(9,16,12,13)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35,40,38)(34,41,43,44)(36,45,39,48)(37,46,42,47);(2,4)(5,10)(9,12)(13,16)(18,25)(19,30)(20,27)(28,32)(33,36,40,39)(34,42,43,37)(35,45,38,48)(41,47,44,46);(1,4,7,2)(3,10,8,5)(6,12,15,9)(11,16,14,13)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,37)(34,36)(35,46)(38,47)(39,43)(40,42)(41,45)(44,48);(1,5,7,10)(2,8,4,3)(6,13,15,16)(9,14,12,11)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,38,40,35)(34,41,43,44)(36,45,39,48)(37,47,42,46);(1,6,7,15)(2,9,4,12)(3,11,8,14)(5,13,10,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,39,40,36)(34,37,43,42)(35,45,38,48)(41,47,44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=g,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, 9150704);