#496201

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403397);