#148587

order	512
nilpotency class	4
isoclinism	isoclinic class of #148586
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11197
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi,g^c=ghi,g^e=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55783);