#9814934

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3811 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)(3,7,9,10)(5,8,14,12)(11,15,13,16)(17,18,23,26)(19,27,29,22)(20,24,21,25)(28,32,31,30)(35,38)(40,42)(43,46)(47,48);(3,9)(7,10)(11,13)(15,16)(17,19)(18,22)(20,28)(21,31)(23,29)(24,30)(25,32)(26,27)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,20,23,21)(18,24,26,25)(19,28,29,31)(22,30,27,32)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,18)(19,22)(20,24)(21,25)(23,26)(27,29)(28,30)(31,32)(33,36,39,34)(35,40,38,42)(37,44,45,41)(43,47,46,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,21,23,20)(18,25,26,24)(19,28,29,31)(22,30,27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,19)(20,30)(21,32)(23,27)(24,28)(25,31)(26,29)(33,38,39,35)(34,40,36,42)(37,43,45,46)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=h,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, 9722130);