#133797

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

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

normal subgroups only

This group has 464 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,e^2=i,f^2=hi,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, 40993);