#335224

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242420);