#190027

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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1141 conjugacy classes of subgroups; only the 284 normal subgroups are shown.

normal subgroups only

This group has 1141 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6,5,7,3,8)(9,10)(11,19,22,15)(12,23)(13,17)(14,18)(16,25)(20,27,34,31)(21,32,35,28)(24,36)(26,30)(29,37)(33,38,40,39);(9,11,14,22)(10,15,18,19)(12,20,25,34)(13,21,26,35)(16,27,23,31)(17,28,30,32)(24,33,37,40)(29,38,36,39);(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,27,19,31)(17,29,30,36)(21,33)(22,34)(26,37)(28,38,32,39)(35,40);(1,3,5,4)(2,6,7,8)(9,13,14,26)(10,17,18,30)(11,21,22,35)(12,24,25,37)(15,28,19,32)(16,29,23,36)(20,33,34,40)(27,38,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 97223);