#125320

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

256 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=fgi,d^2=g,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=efg,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, 32516);