#129357

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8301
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

807 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 807 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,e^c=ei,f^c=fi,f^d=fi

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, 36553);