#133788

order	512
nilpotency class	4
isoclinism	isoclinic class of #133474
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	C₂×#117	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 462 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,e^b=eh,f^b=fg,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, 40984);