#9836094

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	#48210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=gi,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, 9743290);