#9243489

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	#45892	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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,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, 9150685);