#105230

order	512
nilpotency class	3
isoclinism	isoclinic class of #102857
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator, Frattini, cosocle	C₄³	C₂³
socle	C₂³	#399
prime radical	C₂×#127	Q₈
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

367 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fi,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12426);