#125372

order	512
nilpotency class	4
isoclinism	isoclinic class of #125297
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

351 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,13,19,14,20,28,38,15,21,29,17,31,18,12,27)(11,22,24,37,25,16,32,34,26,35,40,39,30,36,23,33);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16,21,36)(12,24,28,40)(13,23,29,32)(14,30,31,25)(17,37,19,39)(18,35,20,22)(27,33,38,34);(2,6)(5,7)(9,12,14,13,15,28,31,29)(10,17,18,38,21,19,20,27)(11,23,25,24,26,32,30,40)(16,33,22,39,36,34,35,37)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32568);