#147296

order	512
nilpotency class	4
isoclinism	isoclinic class of #12933
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1853
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314209, #72314275
maximal normal subgroup of	#72314209, #72314275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54492);