#483142

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,22,28,20,26,35,27,23,25,38,44,36,43,21,37)(19,29,32,39,34,45,48,40,33,46,47,24,30,41,31,42);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,33)(18,24,25,39)(20,30,36,34)(21,32,35,47)(22,31,38,48)(26,40,43,42)(27,45,37,41)(28,46,44,29);(2,7)(6,9)(8,15)(14,16)(18,25)(24,39)(26,43)(27,37)(28,44)(29,46)(40,42)(41,45);(17,20,23,36)(18,26,25,43)(19,30,33,34)(21,22,35,38)(24,40,39,42)(27,44,37,28)(29,41,46,45)(31,47,48,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390338);