#110194

order	512
nilpotency class	3
isoclinism	isoclinic class of #110191
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di

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, 17390);