#401629

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14820
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1751 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:

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

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,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 308825);