#118734

order	512
nilpotency class	3
isoclinism	isoclinic class of #118730
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

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, 25930);