#8093287

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

4568 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:

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

Polycyclic presentation:

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

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, 8000483);