#148895

order	512
nilpotency class	4
isoclinism	isoclinic class of #134702
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11232
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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)(3,15)(4,21,24,52)(5,26)(6,11)(7,31)(8,13)(9,17)(10,23,38,57)(12,28)(14,27)(16,32,47,36)(18,35)(19,41)(20,44)(22,53,55,39)(25,58,54,37)(29,43)(30,40)(33,49)(34,50)(42,46,60,56)(45,51,64,61)(48,62,63,59);(1,3,6,29,7,19,28,50,8,20,30,18,27,49,5,17)(2,9,26,33,14,35,40,44,13,34,12,41,31,43,11,15)(4,16,23,56,25,48,55,64,24,47,57,46,54,63,22,45)(10,36,52,61,39,62,58,42,38,32,21,51,53,59,37,60);(1,4,5,22,27,54,30,57,8,24,28,55,7,25,6,23)(2,10,11,37,31,53,12,21,13,38,40,58,14,39,26,52)(3,16,17,45,49,63,18,46,20,47,50,64,19,48,29,56)(9,32,15,42,43,62,41,61,34,36,44,60,35,59,33,51)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,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, 56091);