#9910132

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

5004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=g,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, 9817328);