#129780

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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^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, 36976);