#128433

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₂²⋊C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fg,d^b=dg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35629);