#190110

order	512
nilpotency class	3
isoclinism	isoclinic class of #190107
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	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1510 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1510 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)(3,11,13,7)(4,15,17,21)(5,9)(6,10)(8,16,22,32)(12,26,28,24)(14,25,30,20)(18,23)(19,27,31,29);(1,3,6,14)(2,7,10,20)(4,12,17,29)(5,13,18,30)(8,19,22,26)(9,11,23,25)(15,31,21,24)(16,27,32,28);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,24)(10,22,23,15)(13,27)(14,29)(18,32)(20,26,25,31)(28,30);(3,13)(7,11)(12,27)(14,30)(19,24)(20,25)(26,31)(28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,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, 97306);