#401968

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14826
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

956 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:

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

Polycyclic presentation:

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

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, 309164);