#190188

order	512
nilpotency class	3
isoclinism	isoclinic class of #190116
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1055 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97384);