#9206647

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,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, 9113843);