#9716234

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	#45489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9623430);