#115659

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1006 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)(3,11)(4,8)(5,14)(6,10)(7,12)(9,15)(13,16)(17,18,21,25,22,24,20,29)(19,26,27,23,28,32,30,31);(1,3,6,4)(2,7,10,16)(5,13,15,12)(8,14,11,9)(17,19,22,28)(18,23,24,31)(20,30,21,27)(25,32,29,26);(1,4,6,3)(2,8,10,11)(5,12,15,13)(7,9,16,14)(18,24)(20,21)(26,32)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 22855);