#362198

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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)(9,10,14,18,15,20,13,19)(11,16,24,32,22,31,23,33)(12,25,28,35,29,36,27,17)(21,34,26,39,37,40,38,30);(2,5)(6,7)(9,11,13,23,15,22,14,24)(10,16,19,33,20,31,18,32)(12,26,28,37,29,38,27,21)(17,34,25,39,35,40,36,30);(9,12,13,27,15,29,14,28)(10,17,18,25,20,35,19,36)(11,21,23,38,22,37,24,26)(16,30,32,34,31,39,33,40);(1,3,4,8)(2,6,5,7)(11,22)(16,31)(21,37)(23,24)(26,38)(30,39)(32,33)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269394);