#325593

order	512
nilpotency class	3
isoclinism	isoclinic class of #278155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=eg,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^c=dh,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, 232789);