#133036

order	512
nilpotency class	4
isoclinism	isoclinic class of #132559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 501 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40232);