#6629535

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2176 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,h^a=hi,e^b=ei,e^c=ei,h^d=hi

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, 6536731);