#190064

order	512
nilpotency class	3
isoclinism	isoclinic class of #190062
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

866 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^b=gh,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97260);