#244443

order	512
nilpotency class	3
isoclinism	isoclinic class of #237997
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2398
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

469 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ef,f^2=h,g^2=hi,b^a=be,c^a=cf,g^a=gh,c^b=cg,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=ghi

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, 151639);