#480171

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
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	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 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)(17,18,22,29,24,28,37,45,23,27,36,46,21,38,20,26)(19,30,33,44,32,25,31,42,34,43,47,40,39,41,35,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,27,41)(20,35,37,31)(21,39,24,32)(22,33,36,47)(26,44,45,40)(28,43,38,30)(29,42,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,36,23,37,24,22)(18,26,38,46,27,45,28,29)(19,31,39,33,34,35,32,47)(25,40,30,42,41,44,43,48);(18,27)(25,41)(26,45)(28,38)(29,46)(30,43)(40,44)(42,48)

Polycyclic presentation:

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

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, 387367);