#493742

order	512
nilpotency class	4
isoclinism	isoclinic class of #480319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1799
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1083 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=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, 400938);