#115176

order	512
nilpotency class	3
isoclinism	isoclinic class of #115163
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1390 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1390 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22372);