#133000

order	512
nilpotency class	4
isoclinism	isoclinic class of #132501
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	C₂⁴	C₂×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

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

normal subgroups only

This group has 473 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:

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

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=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fgi,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, 40196);