#9923993

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gi,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, 9831189);