#9159958

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18)(19,23)(20,25)(21,28)(22,24)(26,32)(27,30)(29,31)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,19,22,27)(18,23,24,30)(20,26,31,28)(21,25,32,29)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,9)(4,15)(8,12)(11,13)(18,24)(19,27)(25,29)(26,28)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,27)(30,32)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,21)(18,26)(19,25)(20,30)(22,32)(23,31)(24,28)(27,29)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067154);