#9945523

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	#55333	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,8,10,5)(4,9,12,15)(11,13,16,14)(18,22)(20,27)(23,29)(25,28)(35,41)(39,45)(42,47)(46,48);(2,7)(3,10)(4,12)(13,14)(18,22)(23,29)(24,30)(31,32)(34,38)(35,41)(40,43)(42,47);(17,18,21,22)(19,23,26,29)(20,24,27,30)(25,31,28,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,13)(10,16)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,5,6,8)(2,3,7,10)(4,13,12,14)(9,16,15,11)(17,20)(18,24)(19,25)(21,27)(22,30)(23,31)(26,28)(29,32)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852719);