#480173

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

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

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,c^a=ci,e^a=eg,d^b=di,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, 387369);