#6817244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=fi,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724440);