#6583032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490228);