#237738

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1168 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 226 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=i,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144934);