#108246

order	512
nilpotency class	3
isoclinism	isoclinic class of #108238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

987 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 987 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,e^b=ei,f^b=fhi,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, 15442);