#127770

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1551
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8044
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=i,f^2=ghi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=ch,d^b=dh,g^b=gi,h^b=hi,d^c=di

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, 34966);