#478547

order	512
nilpotency class	3
isoclinism	isoclinic class of #476143
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2819
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17153
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,f^c=fi,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385743);