#10444993

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

4229 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)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(25,26,31,28)(27,34,32,29)(30,36,38,33)(35,37,39,40);(1,2,5,4)(3,7,8,6)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(25,26,31,28)(27,34,32,29)(30,33,38,36)(35,40,39,37);(1,3,5,8)(2,6,4,7)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(25,28,31,26)(27,34,32,29)(30,33,38,36)(35,37,39,40);(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,30,31,38)(26,33,28,36)(27,35,32,39)(29,37,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10352189);