#413343

order	512
nilpotency class	3
isoclinism	isoclinic class of #412862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2858
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15144
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1169 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)(3,9)(4,19,22,38)(5,25)(6,12)(7,29)(8,13)(10,23,37,24)(11,27)(14,35,41,58)(15,43)(16,32)(17,33)(18,34)(20,48,42,30)(21,36,50,60)(26,39)(28,54)(31,44)(40,61,62,55)(45,57)(46,56)(47,52,53,51)(49,64,63,59);(1,3)(2,9)(4,20,22,42)(5,15)(6,17)(7,16)(8,18)(10,35,37,58)(11,31)(12,33)(13,34)(14,24,41,23)(19,30,38,48)(21,49,50,63)(25,43)(26,45)(27,44)(28,46)(29,32)(36,59,60,64)(39,57)(40,52,62,51)(47,55,53,61)(54,56);(1,4,6,24,8,22,7,23)(2,10,12,38,13,37,29,19)(3,14,16,42,18,41,17,20)(5,21,26,52,28,50,27,51)(9,30,32,58,34,48,33,35)(11,36,25,53,39,60,54,47)(15,40,44,63,46,62,45,49)(31,55,56,64,57,61,43,59);(1,5,7,27,8,28,6,26)(2,11,12,25,13,39,29,54)(3,15,16,44,18,46,17,45)(4,21,23,51,22,50,24,52)(9,31,33,43,34,57,32,56)(10,36,38,53,37,60,19,47)(14,40,42,63,41,62,20,49)(30,55,35,59,48,61,58,64)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=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, 320539);