#500472

order	512
nilpotency class	4
isoclinism	isoclinic class of #500471
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24128
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28181	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

737 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307725, #72307951
maximal normal subgroup of	#72307725, #72307951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407668);