#451196

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15988
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

745 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358392);