#9443234

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

4015 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,27,28,24)(20,25,26,21)(29,32,31,30)(33,34,37,41)(35,38,43,39)(36,40,45,48)(42,46,44,47);(3,9)(7,13)(10,15)(14,16)(18,23)(20,26)(24,27)(29,31)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(2,6)(7,13)(8,11)(14,16)(18,23)(19,28)(21,25)(29,31)(33,35,37,43)(34,39,41,38)(36,42,45,44)(40,47,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,28)(18,24,23,27)(20,31,26,29)(21,30,25,32)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,26)(18,25,23,21)(19,29,28,31)(24,30,27,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,25)(18,26,23,20)(19,30,28,32)(24,29,27,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=g,e^2=g,f^2=g,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, 9350430);