#475231

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17034
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

575 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)(3,13,16,8)(4,9)(5,22,24,10)(6,12)(7,11)(14,27,40,36)(15,28)(17,38,44,29)(18,30,46,57)(19,47,23,31)(20,33)(21,32)(25,50,53,34)(26,35)(37,43)(39,60)(41,54)(42,55)(45,63)(48,64,52,58)(49,59)(51,56)(61,62);(1,3)(2,8)(4,18,20,46)(5,15)(6,17)(7,16)(9,30,33,57)(10,28)(11,13)(12,29)(14,21,40,49)(19,45,23,51)(22,37)(24,43)(25,42)(26,44)(27,32,36,59)(31,56,47,63)(34,55)(35,38)(39,48,41,52)(50,61)(53,62)(54,58,60,64);(1,4)(2,9)(3,14,16,40)(5,23,24,19)(6,21)(7,20)(8,27,13,36)(10,31,22,47)(11,33)(12,32)(15,41)(17,18,44,46)(25,52,53,48)(26,49)(28,54)(29,30,38,57)(34,58,50,64)(35,59)(37,60)(39,43)(42,51)(45,62)(55,56)(61,63);(1,5,6,25,7,24,26,53)(2,10,12,34,11,22,35,50)(3,15,17,42,16,43,44,62)(4,19,21,48,20,23,49,52)(8,28,29,55,13,37,38,61)(9,31,32,58,33,47,59,64)(14,39,18,45,40,41,46,51)(27,54,30,56,36,60,57,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382427);