#148322

order	512
nilpotency class	4
isoclinism	isoclinic class of #134701
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#11096
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,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, 55518);