#436681

order	512
nilpotency class	3
isoclinism	isoclinic class of #436680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15567
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1125 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)(3,15,19,9)(4,20,23,39)(5,25,27,38)(6,13)(7,14)(8,12)(10,24,35,29)(11,28,37,26)(16,42)(17,46)(18,40,30,33)(21,49)(22,52)(31,44)(32,47)(34,53)(36,57)(41,45)(43,59,51,58)(48,55)(50,56)(54,60)(61,63,64,62);(1,3,6,18,8,19,7,30)(2,9,13,33,12,15,14,40)(4,21,24,45,23,16,29,44)(5,17,26,47,27,48,28,57)(10,31,20,49,35,41,39,42)(11,36,38,46,37,32,25,55)(22,43,54,62,53,51,56,63)(34,58,50,64,52,59,60,61);(1,4,8,23)(2,10,12,35)(3,16,19,21)(5,22,27,53)(6,29,7,24)(9,31,15,41)(11,34,37,52)(13,39,14,20)(17,43,48,51)(18,45,30,44)(25,50,38,60)(26,56,28,54)(32,58,36,59)(33,42,40,49)(46,64,55,61)(47,63,57,62);(1,5,6,26,8,27,7,28)(2,11,14,25,12,37,13,38)(3,17,18,47,19,48,30,57)(4,22,29,56,23,53,24,54)(9,32,40,46,15,36,33,55)(10,34,20,50,35,52,39,60)(16,43,45,63,21,51,44,62)(31,58,49,64,41,59,42,61)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318217, #72318894, #72318981
maximal normal subgroup of	#72318217, #72318894, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343877);