#129778

order	512
nilpotency class	4
isoclinism	isoclinic class of #129777
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8328
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60462	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

852 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,14,18,42)(4,20,23,37)(5,25,28,57)(6,12)(7,30)(8,13)(9,16)(10,22,36,24)(11,27,39,29)(15,43)(17,41,19,38)(21,35)(26,32)(31,58,59,52)(33,48)(34,49)(40,46)(44,50,64,60)(45,62)(47,63)(51,53)(54,56)(55,61);(1,3)(2,9)(4,15)(5,26)(6,17)(7,19)(8,18)(10,31)(11,38)(12,32)(13,34)(14,25)(16,29)(20,50)(21,52)(22,45)(23,47)(24,46)(27,49)(28,48)(30,33)(35,43)(36,59)(37,60)(39,41)(40,56)(42,57)(44,53)(51,62)(54,64)(55,58)(61,63);(1,4,8,23)(2,10,13,36)(3,15,18,47)(5,21,28,55)(6,24,7,22)(9,31,34,59)(11,35,39,61)(12,37,30,20)(14,40,42,62)(16,44,49,64)(17,46,19,45)(25,56,57,51)(26,52,48,58)(27,54,29,53)(32,60,33,50)(38,43,41,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36974);