#499369

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2073
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320615, #72320647, #72320752
maximal normal subgroup of	#72320615, #72320647, #72320752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406565);