#7500920

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500919
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2669
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3069 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:

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

Polycyclic presentation:

2^9|a^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fi,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	#72264928, #72264949, #72265007, …
maximal normal subgroup of	#72264928, #72264949, #72265007, …
maximal quotient of	#72265688

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408116);