#148382

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#11121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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)(3,16,21,48)(4,23,28,44)(5,14)(6,13)(7,32)(8,34)(9,15)(10,18,38,55)(11,26,42,29)(12,30)(17,49)(19,45,22,37)(20,36,33,60)(24,57)(25,46,58,40)(27,43,59,41)(31,47)(35,56)(39,52)(50,61)(51,63)(53,64)(54,62);(1,3,5,18,8,22,30,20,9,21,31,55,6,19,7,33)(2,10,12,16,13,36,14,45,15,38,32,48,34,60,47,37)(4,24,25,53,29,56,59,51,28,54,58,17,26,50,27,52)(11,39,40,35,23,49,41,57,42,63,46,61,44,64,43,62);(1,4)(2,11)(3,17)(5,27)(6,29)(7,25)(8,26)(9,28)(10,35)(12,43)(13,44)(14,46)(15,42)(16,39)(18,54)(19,52)(20,56)(21,53)(22,51)(23,34)(24,55)(30,58)(31,59)(32,41)(33,50)(36,62)(37,49)(38,61)(40,47)(45,64)(48,63)(57,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dhi,e^c=ei,f^c=fh,g^c=gi,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, 55578);