#133804

order	512
nilpotency class	4
isoclinism	isoclinic class of #133422
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, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2312 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 488 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,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,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, 41000);