#8093104

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4658 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000300);