#9726070

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	#55356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633266);