#115655

order	512
nilpotency class	3
isoclinism	isoclinic class of #115640
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1114 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,7)(4,6)(5,9)(8,12)(10,15)(11,16)(13,14)(17,18,21,25,22,24,20,29)(19,26,27,23,28,32,30,31);(2,6)(4,12)(7,15)(8,9)(10,13)(14,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,31)(25,32)(26,29);(1,3,5,11)(2,7,9,16)(4,13,12,10)(6,14,8,15)(18,24)(20,21)(26,32)(27,30)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,d^c=dh,g^c=gi

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, 22851);