#362968

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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:

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

Polycyclic presentation:

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

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, 270164);