#9863369

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,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, 9770565);