#455199

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16261
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1252 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)(3,15)(4,21)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30,40,29)(16,44)(17,48,51,33)(19,35)(20,34)(22,32)(23,55,56,36)(24,38)(26,37)(28,39,57,53)(42,47)(43,52,62,50)(45,61,54,63)(46,60)(49,59,58,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,47,26,16,25,46)(5,17,29,50,31,51,30,52)(10,32,37,60,38,44,21,42)(11,33,27,43,40,48,41,62)(23,54,28,58,56,45,57,49)(36,59,39,61,55,64,53,63);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,39)(12,21)(13,37)(14,38)(15,42)(17,49)(18,47)(19,46)(20,22)(23,29)(27,36)(30,56)(31,57)(33,61)(34,60)(35,44)(40,53)(41,55)(43,59)(45,50)(48,63)(51,58)(52,54)(62,64);(1,5,6,29,8,31,7,30)(2,11,12,27,14,40,13,41)(3,17,18,50,20,51,19,52)(4,23,24,28,26,56,25,57)(9,33,15,43,35,48,34,62)(10,36,37,39,38,55,21,53)(16,45,46,49,22,54,47,58)(32,59,60,61,44,64,42,63)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,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, 362395);