#319339

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

866 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=egi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226535);