#480166

order	512
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

737 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,27,23,28,36,25,22,26,38,43,35,42,20,34)(19,29,32,44,33,46,30,39,31,24,37,40,47,41,48,45);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,31)(18,24,26,29)(20,30,36,48)(21,37,38,32)(23,33,35,47)(25,39,34,45)(27,44,43,40)(28,41,42,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,35,38,22,36,23,21)(18,25,42,27,26,34,28,43)(19,30,47,32,31,48,33,37)(24,39,46,44,29,45,41,40);(19,31)(24,29)(30,48)(32,37)(33,47)(39,45)(40,44)(41,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,f^2=h,h^2=i,b^a=be,c^a=ci,e^a=eg,d^b=di,f^b=fg

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, 387362);