#495411

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=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, 402607);