#132113

order	512
nilpotency class	4
isoclinism	isoclinic class of #131997
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

607 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39309);