#6533262

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3320 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317675, #72319333
maximal normal subgroup of	#72317675, #72319333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440458);