#121987

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1446
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

943 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 943 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,g^b=gh,d^c=dh,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29183);