#7500764

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500763
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2668
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2667 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306392, #72306453, #72306460
maximal normal subgroup of	#72306392, #72306453, #72306460

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407960);