#9850978

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	#46040	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9758174);