#146753

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

939 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)(3,15,18,33)(4,21)(5,26)(6,12)(7,14)(8,17,34,53)(9,22)(10,29)(11,42)(13,31)(16,48,50,32)(19,43,20,39)(23,40)(24,44)(25,36)(27,45,54,35)(28,37)(30,41)(38,58)(46,57)(47,63,56,49)(51,62,55,61)(52,59,64,60);(1,3,6,19,7,18,31,20)(2,8,12,35,14,34,13,45)(4,16,23,51,25,50,24,55)(5,27,29,53,30,54,42,17)(9,32,37,61,38,48,46,62)(10,39,41,15,11,43,26,33)(21,56,40,59,36,47,44,60)(22,52,28,49,58,64,57,63);(1,4)(2,9)(3,16)(5,28)(6,23)(7,25)(8,32)(10,40)(11,44)(12,37)(13,46)(14,38)(15,47)(17,52)(18,50)(19,51)(20,55)(21,26)(22,42)(24,31)(27,49)(29,58)(30,57)(33,56)(34,48)(35,61)(36,41)(39,59)(43,60)(45,62)(53,64)(54,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53949);