#146904

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10807
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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)(3,14,18,33)(4,19,22,54)(5,10)(6,26)(7,13)(8,16,32,49)(9,21,35,56)(11,24)(12,25)(15,43)(17,31,27,38)(20,53,23,39)(28,36,57,34)(29,37)(30,59)(40,50)(41,55)(42,58,63,48)(44,61)(45,60)(46,52)(47,64)(51,62);(1,3,5,16,7,18,25,49)(2,8,12,33,13,32,10,14)(4,15,23,46,22,47,20,44)(6,27,24,58,29,17,50,48)(9,30,34,60,35,62,36,41)(11,38,26,42,40,31,37,63)(19,52,53,43,54,61,39,64)(21,55,57,51,56,45,28,59);(1,4,7,22)(2,9)(3,15,18,47)(5,23,25,20)(6,28,29,57)(8,30)(10,36)(11,39)(12,34)(13,35)(14,41)(16,46,49,44)(17,51,27,59)(19,26)(21,50,56,24)(31,43)(32,62)(33,60)(37,54)(38,64)(40,53)(42,52)(45,58,55,48)(61,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,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, 54100);