#148321

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

659 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)(3,13)(4,19,21,9)(5,23)(6,26)(7,11)(8,16)(10,14)(12,24)(15,34,42,57)(17,36)(18,46)(20,49,51,52)(22,32,50,48)(25,53)(27,54,30,41)(28,45)(29,56)(31,39,47,61)(33,40)(35,38)(37,44)(43,63,62,59)(55,60,58,64);(1,3,14,38,6,18,25,45,7,17,40,56,24,44,5,16)(2,8,23,37,12,29,33,36,11,28,53,46,26,35,10,13)(4,15,39,60,22,43,51,54,21,42,61,64,50,62,20,41)(9,30,52,63,32,58,47,34,19,27,49,59,48,55,31,57);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,27)(10,31)(11,19)(12,32)(13,34)(14,39)(16,41)(17,42)(18,43)(23,52)(24,50)(25,51)(26,48)(28,30)(29,55)(33,47)(35,58)(36,57)(37,59)(38,60)(40,61)(44,62)(45,54)(46,63)(49,53)(56,64)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,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, 55517);