#8455720

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

4375 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8362916);