#142292

order	512
nilpotency class	4
isoclinism	isoclinic class of #142242
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33520	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

588 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,5)(4,6)(7,8)(10,15)(11,20)(12,24)(13,26)(16,31)(17,25)(18,29)(19,30)(21,36)(22,38)(23,35)(27,39)(28,32)(33,37);(5,8)(9,10)(11,21)(12,25)(13,17)(14,19)(15,24)(16,23)(18,26)(20,33)(22,31)(27,29)(28,40)(30,39)(32,35)(34,37)(36,38);(1,3)(2,5)(4,7)(6,8)(9,11,13,20,14,23,26,35)(10,16,18,33,19,21,17,32)(12,22,24,34,27,38,39,40)(15,28,25,36,30,37,29,31)

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,g^a=gi,c^b=cf,d^b=dg,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49488);