#6668618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,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,g^c=gi,h^c=hi,e^d=ei,h^e=hi

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, 6575814);