#190151

order	512
nilpotency class	3
isoclinism	isoclinic class of #190116
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1050 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1050 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,g^2=h,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, 97347);