#132990

order	512
nilpotency class	4
isoclinism	isoclinic class of #132509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40186);