#133796

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 446 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,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, 40992);