#470588

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16690
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 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:

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

Polycyclic presentation:

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

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, 377784);