#190186

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	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1354 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=e,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, 97382);