#430122

order	512
nilpotency class	3
isoclinism	isoclinic class of #427241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15413
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

934 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:

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

Polycyclic presentation:

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

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, 337318);