#9206579

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

3801 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=gh,e^2=hi,f^2=h,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, 9113775);