#132171

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	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 600 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,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, 39367);