#125392

order	512
nilpotency class	4
isoclinism	isoclinic class of #125300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

255 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^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,g^b=gi,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, 32588);