#320211

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
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	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

660 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 190 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)(9,10,12,15,13,16,21,24)(11,17,18,22,19,23,20,14)(25,26,29,33,28,32,35,31)(27,34,38,40,37,30,36,39);(2,5)(6,7)(25,27,35,36,28,37,29,38)(26,30,31,40,32,34,33,39);(9,11,13,19)(10,14,16,22)(12,20,21,18)(15,23,24,17)(26,31)(27,36)(29,35)(30,34)(32,33)(37,38);(1,3,4,8)(2,6,5,7)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227407);