#129137

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#106
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	C₈×#106
prime radical	C₂³	#443
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,13,16,32,17,39,24,51)(4,9)(5,10)(6,12)(7,11)(8,15,27,21,28,43,36,49)(14,38,41,54,42,60,47,64)(18,29)(19,31)(20,30)(22,34)(23,33)(25,35)(26,40,52,44,53,61,58,62)(37,50)(45,56)(46,55)(48,57)(59,63);(1,3)(2,8,11,28)(4,14)(5,21)(6,24)(7,17)(9,26,30,53)(10,32,33,51)(12,36,35,27)(13,37,39,34)(15,22)(16,25)(18,44)(19,47)(20,42)(23,49)(29,54,55,64)(31,58,57,52)(38,59,60,56)(40,45)(41,48)(43,50)(46,62)(61,63);(1,4,5,18,6,19,22,45,7,20,23,46,25,48,50,63)(2,9,10,29,12,31,34,56,11,30,33,55,35,57,37,59)(3,14,21,44,24,47,15,40,17,42,49,62,16,41,43,61)(8,26,32,54,36,58,13,38,28,53,51,64,27,52,39,60)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,g^2=i,b^a=bd,f^a=fi,d^b=dg,e^b=egi,g^b=gi,h^b=hi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320434, #72320727, #72320780
maximal normal subgroup of	#72320434, #72320727, #72320780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36333);