#146898

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54094);