#9794261

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	#51977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9701457);