#363558

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1000 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)(11,18)(12,17,27,35)(13,28)(14,30)(15,20)(16,22)(19,29)(21,33,24,39)(23,36)(25,32,38,31)(26,40,37,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,22,15,23,29,13)(10,16,19,18,20,28,30,36)(12,24,26,25,27,21,37,38)(17,32,34,39,35,31,40,33);(2,5)(6,8)(9,12,14,26,15,27,29,37)(10,17,19,34,20,35,30,40)(11,21,22,38,23,24,13,25)(16,31,18,33,28,32,36,39);(1,3,4,7)(2,6,5,8)(9,13,29,23,15,22,14,11)(10,18,30,16,20,36,19,28)(12,25,37,24,27,38,26,21)(17,33,40,31,35,39,34,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,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, 270754);