#356005

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1554 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)(12,16,22,29)(13,24)(14,17)(18,25)(19,26,31,37)(20,27)(21,34)(23,36,35,30)(28,33)(32,40,39,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,27)(18,28)(22,31)(23,32)(24,34)(25,33)(29,37)(30,38)(35,39)(36,40);(3,8)(5,7)(9,12,13,23,14,22,25,35)(10,16,18,30,17,29,24,36)(11,19,21,32,20,31,33,39)(15,26,28,38,27,37,34,40);(2,6)(5,7)(11,20)(15,27)(19,31)(21,33)(26,37)(28,34)(32,39)(38,40)

Polycyclic presentation:

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

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, 263201);