#190190

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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1009 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1009 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:

(1,2,4,8,5,6,11,14)(3,7,10,15,9,13,12,16)(17,18,21,26,22,25,20,24)(19,27,28,32,29,23,30,31);(2,6)(4,11)(7,13)(10,12);(1,3)(2,7,6,13)(4,12)(5,9)(8,16,14,15)(10,11)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(3,9)(7,13)(10,12)(15,16)(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=hi,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, 97386);