#149175

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11316
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

405 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:

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

Polycyclic presentation:

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

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, 56371);