#118733

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	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

530 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,g^2=h,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, 25929);