#499515

order	512
nilpotency class	4
isoclinism	isoclinic class of #499514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23708
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320520, #72320565, #72320582
maximal normal subgroup of	#72320520, #72320565, #72320582
maximal quotient of	#72321352, #72321547, #72322371, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406711);