#149359

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11372
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

427 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56555);