#133776

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

608 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,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, 40972);