#142241

order	512
nilpotency class	4
isoclinism	isoclinic class of #142240
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	#23386	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

808 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,37)(12,23,25,39)(13,28,26,17)(15,30,31,20)(16,33,35,27)(22,38,29,34)(24,36,40,32);(5,8)(9,11)(10,15)(12,24)(13,29)(14,21)(16,34)(17,36)(18,31)(19,23)(20,27)(22,26)(25,40)(28,32)(30,33)(35,38)(37,39);(1,3)(2,5)(4,7)(6,8)(12,25)(16,35)(19,37)(20,30)(23,39)(24,40)(27,33)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49437);