#184394

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

672 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 366 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)(4,8)(5,6)(9,10,12,16,13,14,20,21)(11,17,18,24,19,15,22,23)(25,26,29,33)(27,35,36,31)(28,32,37,40)(30,34,38,39);(2,6)(7,8)(25,27,30,37,29,36,38,28)(26,31,34,32,33,35,39,40);(1,3,5,4)(2,7,6,8)(10,14)(15,17)(16,21)(23,24)(25,28,38,36,29,37,30,27)(26,32,39,31,33,40,34,35);(1,4,5,3)(2,8,6,7)(9,11,12,18,13,19,20,22)(10,15,16,23,14,17,21,24)(26,33)(31,35)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91590);