#190026

order	512
nilpotency class	3
isoclinism	isoclinic class of #190022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1337 conjugacy classes of subgroups; only the 284 normal subgroups are shown.

normal subgroups only

This group has 1337 conjugacy classes of subgroups — too many to draw. Only its 284 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,g^b=gh,f^c=fh

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, 97222);