#345531

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

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, 252727);