#8093284

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

4650 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)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(25,26,30,33)(27,34,37,31)(28,32,38,29)(35,36,40,39);(3,8)(4,7)(10,15)(12,17)(16,21)(20,23)(28,38)(29,32)(35,40)(36,39);(1,3)(2,4)(5,8)(6,7)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,28)(26,32)(27,35)(29,33)(30,38)(31,39)(34,36)(37,40);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,29,30,32)(26,28,33,38)(27,36,37,39)(31,35,34,40);(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40)

Polycyclic presentation:

2^9|a^2=gh,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, 8000480);