#8095647

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

4508 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002843);