#6629538

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	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2192 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)(11,22)(12,17)(13,27)(14,20)(15,31)(16,21)(18,24)(19,28)(23,32)(25,34)(26,33)(29,36)(30,35)(37,38)(39,40);(2,5)(6,8)(9,11,14,26,16,24,15,12)(10,17,31,18,21,33,20,22)(13,23,28,38,30,39,29,25)(19,32,27,34,36,40,35,37);(1,3,4,7)(2,6,5,8)(9,12,15,24,16,26,14,11)(10,18,20,17,21,22,31,33)(13,25,29,39,30,38,28,23)(19,34,35,32,36,37,27,40);(9,13)(10,19)(11,23)(12,25)(14,28)(15,29)(16,30)(17,32)(18,34)(20,35)(21,36)(22,37)(24,39)(26,38)(27,31)(33,40);(9,14,16,15)(10,20,21,31)(11,12,24,26)(13,28,30,29)(17,18,33,22)(19,35,36,27)(23,25,39,38)(32,34,40,37)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,e^2=i,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, 6536734);