#9236996

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	#41981	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4472 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)(11,20)(12,21)(13,19)(14,24)(15,17)(16,22)(18,23)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11,15,22)(10,16,17,20)(12,13,23,24)(14,18,19,21)(27,33)(30,37)(34,39)(38,40);(2,5)(6,7)(10,17)(12,23)(13,24)(16,20)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(9,12)(10,18)(11,13)(14,20)(15,23)(16,19)(17,21)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,12)(14,17)(15,24)(16,18)(20,21)(22,23)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,3)(2,6)(4,8)(5,7)(9,14,15,19)(10,13,17,24)(11,21,22,18)(12,20,23,16)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9144192);