#243922

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

667 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151118);