#128464

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
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	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₈×#99
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 327 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fgi,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	#72343150, #72343179, #72343185
maximal normal subgroup of	#72343150, #72343179, #72343185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35660);