#440844

order	512
nilpotency class	3
isoclinism	isoclinic class of #440843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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)(4,20,24,38)(5,25)(6,13)(7,14)(8,12)(10,23,37,30)(11,27)(16,43)(17,47,50,57)(18,34,29,33)(21,42)(22,53,56,36)(26,40)(28,41)(31,45)(32,49,39,48)(35,46)(44,58)(51,54,61,55)(52,60)(59,63)(62,64);(1,3,7,18,8,19,6,29)(2,9,14,33,12,15,13,34)(4,21,23,46,24,45,30,16)(5,17,27,48,28,50,26,49)(10,35,20,31,37,43,38,42)(11,39,41,47,40,32,25,57)(22,52,54,64,56,63,55,44)(36,60,51,62,53,59,61,58);(1,4,8,24)(2,10)(3,16,19,46)(5,22,28,56)(6,23,7,30)(9,31)(11,36)(12,37)(13,20)(14,38)(15,42)(17,44,50,64)(18,45,29,21)(25,51)(26,54,27,55)(32,58)(33,35)(34,43)(39,62)(40,53)(41,61)(47,60)(48,63,49,52)(57,59);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,28)(9,32)(10,36)(12,40)(13,25)(14,41)(15,39)(16,44)(18,48)(19,50)(20,51)(21,52)(23,54)(24,56)(29,49)(30,55)(31,58)(33,57)(34,47)(35,59)(37,53)(38,61)(42,62)(43,60)(45,63)(46,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264921, #72264954, #72264996
maximal normal subgroup of	#72264921, #72264954, #72264996
maximal quotient of	#72265588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348040);