#147300

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
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

509 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,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, 54496);