#124118

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#329
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	C₂×#329
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=i,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31314);