#498339

order	512
nilpotency class	4
isoclinism	isoclinic class of #498338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23001
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33594	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

606 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285521, #72285570, #72285871
maximal normal subgroup of	#72285521, #72285570, #72285871
maximal quotient of	#72287433, #72287512, #72287601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405535);