#8467172

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

4572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=h,e^2=hi,f^2=i,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, 8374368);