#142294

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	#33520	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 639 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(12,13)(16,19)(17,21)(18,24)(22,23);(5,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,22)(23,24);(1,3)(2,5)(4,7)(6,8)(9,11,12,20,14,23,13,22)(10,16,18,24,19,15,17,21)

Polycyclic presentation:

2^9|c^2=g,d^2=gi,e^2=i,f^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,f^b=fi,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, 49490);