#481155

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=gi,f^2=h,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=egi,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388351);