#148995

order	512
nilpotency class	4
isoclinism	isoclinic class of #13374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2023
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11267
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

465 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^c=fi,g^c=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, 56191);