#133817

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

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

normal subgroups only

This group has 484 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,e^b=eh,f^b=fg,d^c=dh,e^c=ei,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, 41013);