#147328

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1898
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#10912
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gh,d^b=dh,h^b=hi,e^c=ei,g^d=gi

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, 54524);