#132127

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

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

normal subgroups only

This group has 465 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=hi,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, 39323);