#9243203

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	#45457	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^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, 9150399);