#9668291

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3743 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)(17,18,23,20)(19,24,29,27)(21,25,28,30)(22,26,32,31)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(34,40)(35,43)(41,42)(44,45);(2,9)(3,13)(4,11)(10,16)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20,23,18)(19,27,29,24)(21,25,28,30)(22,26,32,31)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21,23,28)(18,25,20,30)(19,22,29,32)(24,26,27,31)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575487);