#9923996

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=i,e^2=gh,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, 9831192);