#132967

order	512
nilpotency class	4
isoclinism	isoclinic class of #132521
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

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

normal subgroups only

This group has 496 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,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, 40163);