#325957

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

496 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,c^2=fh,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233153);