#148334

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
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

653 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)(3,14)(4,20,24,10)(5,25)(6,13)(7,30)(8,12)(9,17)(11,15)(16,46,49,40)(18,42)(19,41)(21,54,50,62)(22,56,47,57)(23,38,29,37)(26,39)(27,58)(28,43)(31,48,61,64)(32,51)(33,44)(34,59)(35,55,52,63)(36,45,53,60);(1,3,15,44,7,18,27,51,8,19,26,59,6,28,5,17)(2,9,25,43,13,34,39,41,12,32,58,42,30,33,11,14)(4,21,45,48,23,16,47,63,24,50,60,64,29,49,22,55)(10,35,57,46,38,61,53,54,20,52,56,40,37,31,36,62);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,31)(11,36)(12,20)(13,38)(14,40)(15,45)(17,48)(18,50)(19,49)(21,28)(25,57)(26,60)(27,47)(30,37)(32,61)(33,52)(34,35)(39,53)(41,46)(42,54)(43,62)(44,63)(51,64)(55,59)(56,58)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,f^b=fi,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, 55530);