#133808

order	512
nilpotency class	4
isoclinism	isoclinic class of #133434
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 455 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,e^c=ei,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, 41004);