#110100

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
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, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

689 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 174 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh

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, 17296);