#9668296

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	#52383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=hi,e^2=ghi,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, 9575492);