#132998

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

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

normal subgroups only

This group has 491 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 40194);