#289413

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

567 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 567 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196609);