#9206400

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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)(3,5)(4,7)(6,12)(8,15)(9,10)(11,14)(13,16)(17,18)(19,21)(20,29)(22,25)(23,26)(24,28)(27,30)(31,32)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,13)(12,14)(15,16)(17,19)(18,24)(20,27)(21,26)(22,32)(23,28)(25,30)(29,31)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(2,4)(3,9)(8,12)(11,16)(17,20,23,22)(18,25,26,29)(19,27,28,32)(21,31,24,30)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,4,7,2)(3,10,9,5)(6,12,15,8)(11,14,16,13)(17,21,23,24)(18,19,26,28)(20,30,22,31)(25,32,29,27)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,5,7,10)(2,3,4,9)(6,14,15,13)(8,16,12,11)(17,22,23,20)(18,25,26,29)(19,27,28,32)(21,30,24,31)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,6,7,15)(2,8,4,12)(3,11,9,16)(5,13,10,14)(17,19)(18,24)(20,27)(21,26)(22,32)(23,28)(25,30)(29,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=h,e^2=gi,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, 9113596);