#9863078

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

4306 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=i,e^2=ghi,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, 9770274);