#132137

order	512
nilpotency class	4
isoclinism	isoclinic class of #132019
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

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

normal subgroups only

This group has 468 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,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, 39333);