#499390

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2077
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

669 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320565, #72320643, #72320677
maximal normal subgroup of	#72320565, #72320643, #72320677
maximal quotient of	#72322864

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406586);