#243907

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
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

695 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151103);