#132120

order	512
nilpotency class	4
isoclinism	isoclinic class of #132007
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
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 ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 463 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,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, 39316);