#325368

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

548 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 166 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)(3,7)(4,6)(5,8)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27);(1,3,5,4)(2,6,8,7)(9,10,13,14,12,16,11,15);(1,3,5,4)(2,7,8,6)(10,14)(11,13)(15,16)(17,19,20,27,21,26,29,30)(18,22,23,25,24,31,28,32);(1,4,5,3)(2,7,8,6)(19,26)(22,31)(25,32)(27,30)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=ef,d^2=e,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, 232564);