#152361

order	512
nilpotency class	5
isoclinism	isoclinic class of #149797
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,6)(4,5)(7,8)(9,10,12,18,14,19,25,33)(11,15,21,28,20,29,34,40)(13,16,23,30,26,17,24,32)(22,27,37,38,35,31,36,39);(2,5)(6,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,27)(17,31)(18,28)(19,29)(23,37)(24,36)(25,34)(26,35)(30,38)(32,39)(33,40);(1,3,4,7)(2,6,5,8)(10,16)(11,20)(12,23)(13,26)(15,27)(17,19)(21,36)(24,25)(29,31)(30,32)(34,37)(38,39)

Polycyclic presentation:

2^9|a^2=f,c^2=d,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59557);