#127409

order	512
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	#405	C₂³
socle	C₂	#7986
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,10,7,12,22,36,8,13,23,37,30,39,6,29)(3,14,17,44,19,45,18,53,20,46,15,42,47,62,48,63)(5,24,25,56,27,41,26,35,28,57,21,54,55,11,38,40)(9,31,32,50,33,59,60,51,34,52,61,64,43,49,58,16);(1,3,6,18,30,47,23,17,8,20,22,48,7,19,4,15)(2,9,10,32,12,33,36,60,13,34,37,61,39,43,29,58)(5,16,26,49,55,64,25,52,28,51,38,59,27,50,21,31)(11,14,35,42,24,45,54,63,41,46,40,44,57,62,56,53);(1,4,7,22,8,23,30,6)(2,10,12,36,13,37,39,29)(3,15,19,48,20,17,47,18)(5,21,27,38,28,25,55,26)(9,32,33,60,34,61,43,58)(11,35,24,54,41,40,57,56)(14,42,45,63,46,44,62,53)(16,31,50,59,51,52,64,49)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=ghi,e^2=g,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,e^b=eh,e^d=ei

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, 34605);