#8649028

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

4305 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,16,22,19)(12,13,18,17)(20,23,24,21)(25,26,31,28)(27,32,35,29)(30,33,38,36)(34,39,40,37);(2,6)(3,7)(9,10,14,15)(11,19,22,16)(12,13,18,17)(20,21,24,23)(25,27)(26,32)(28,29)(30,34)(31,35)(33,39)(36,37)(38,40);(1,3,5,7)(2,4,6,8)(10,15)(12,18)(16,19)(20,24)(25,26,31,28)(27,29,35,32)(30,33,38,36)(34,37,40,39);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,20)(13,23)(14,22)(15,19)(17,21)(18,24)(25,28,31,26)(27,29,35,32)(30,33,38,36)(34,39,40,37);(9,12,14,18)(10,17,15,13)(11,20,22,24)(16,21,19,23)(25,29)(26,27)(28,35)(30,37)(31,32)(33,34)(36,40)(38,39);(9,13,14,17)(10,18,15,12)(11,21,22,23)(16,20,19,24)(25,30,31,38)(26,33,28,36)(27,34,35,40)(29,37,32,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8556224);