#6487524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4104 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)(3,6)(4,7)(5,8)(9,10,15,18)(11,21,23,35)(12,25,28,36)(13,20,14,19)(16,29,32,24)(17,30,34,27)(22,31,26,33)(37,38,40,39);(3,4)(6,7)(9,11)(10,16)(12,26)(13,29)(14,24)(15,23)(17,33)(18,32)(19,35)(20,21)(22,28)(25,37)(27,38)(30,39)(31,34)(36,40);(1,3,5,4)(2,6,8,7)(9,12,15,28)(10,17,18,34)(11,22,23,26)(13,27,14,30)(16,31,32,33)(19,25,20,36)(21,37,35,40)(24,38,29,39);(1,4,5,3)(2,7,8,6)(10,18)(11,23)(17,34)(19,20)(22,26)(24,29)(25,36)(38,39);(2,8)(6,7)(10,18)(16,32)(17,34)(19,20)(21,35)(25,36)(31,33)(37,40)

Polycyclic presentation:

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

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, 6394720);