#9912945

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	#45449	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4261 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820141);