#9206599

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3746 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113795);