#148012

order	512
nilpotency class	4
isoclinism	isoclinic class of #148011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11043
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

770 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=ei,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, 55208);