#128462

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₈×#99
prime radical	#525	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

329 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319273, #72343185
maximal normal subgroup of	#72319273, #72343185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35658);