#294986

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

759 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,16,21,17)(12,23,22,24)(13,19,14,18)(28,36)(30,38)(32,37)(35,40);(2,5)(6,7)(9,11,15,21)(10,16,20,17)(12,13,22,14)(18,23,19,24)(25,26)(27,33)(28,37)(29,31)(30,35)(32,36)(34,39)(38,40);(9,12,14,11,15,22,13,21)(10,17,19,24,20,16,18,23)(25,27)(26,30,31,38)(28,35)(29,34)(32,33,37,39)(36,40);(1,3,4,8)(2,6,5,7)(11,21)(12,22)(16,17)(23,24)(27,34)(30,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202182);