#6487533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3960 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)(4,7)(5,8)(9,10,16,21)(11,22,17,31)(12,25,27,24)(13,18,28,33)(14,20,15,19)(23,36,32,40)(26,38,39,37)(29,35,30,34);(3,4)(6,7)(9,11,16,17)(10,12,21,27)(13,23,28,32)(14,25,15,24)(18,26,33,39)(19,31,20,22)(29,38,30,37)(34,40,35,36);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,21)(13,26)(14,22)(15,31)(16,27)(18,32)(19,25)(20,24)(23,33)(28,39)(29,36)(30,40)(34,38)(35,37);(1,4,5,3)(2,7,8,6)(9,13,16,28)(10,18,21,33)(11,23,17,32)(12,26,27,39)(14,29,15,30)(19,34,20,35)(22,36,31,40)(24,37,25,38);(2,8)(6,7)(13,28)(18,33)(23,32)(26,39)(29,30)(34,35)(36,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394729);