#352822

order	512
nilpotency class	3
isoclinism	isoclinic class of #352122
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,21,24)(12,16,22,23)(13,20,15,19)(27,33)(28,35)(30,36)(32,38)(34,39)(37,40);(2,5)(6,8)(9,11,13,12,14,21,15,22)(10,16,19,18,17,23,20,24)(25,26,29,31)(27,30,34,37)(28,32,35,38)(33,36,39,40);(10,17)(11,21)(12,22)(19,20)(25,27,28,33,29,34,35,39)(26,30,32,36,31,37,38,40);(1,3,4,7)(2,6,5,8)(9,12,13,21,14,22,15,11)(10,18,19,23,17,24,20,16)(26,31)(30,37)(32,38)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dhi,d^c=dh

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, 260018);