#133809

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2313 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 471 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,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, 41005);