#9706345

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	#38430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 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)(3,8)(4,6)(5,7)(9,10,14,15)(11,19,22,16)(12,17,18,13)(20,21,24,23)(25,26,30,28)(27,31,35,33)(29,32,37,36)(34,38,40,39);(1,3,5,4)(2,6,7,8)(10,15)(12,18)(16,19)(20,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,11,14,22)(10,16,15,19)(12,20,18,24)(13,21,17,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,20)(13,15)(14,18)(16,23)(19,21)(22,24)(25,28,30,26)(27,33,35,31)(29,32,37,36)(34,38,40,39);(1,2)(3,6)(4,8)(5,7)(9,13,14,17)(10,18,15,12)(11,21,22,23)(16,24,19,20)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9613541);