#499744

order	512
nilpotency class	4
isoclinism	isoclinic class of #499742
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23848
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1256 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)(3,13)(4,18)(5,22)(6,28)(7,11)(8,15)(9,19)(10,26)(12,25)(14,32)(16,41)(17,48)(20,54)(21,51)(23,39)(24,40)(27,50)(29,36)(30,37)(31,47)(33,55)(34,53)(35,56)(38,52)(42,43)(44,57)(45,60)(46,58)(49,59)(61,63)(62,64);(1,3)(2,8)(4,14)(5,23)(6,29)(7,16)(9,32)(10,36)(11,31)(12,39)(13,22)(15,26)(17,25)(18,42)(19,46)(20,49)(21,44)(24,43)(27,55)(28,33)(30,45)(34,57)(35,60)(37,58)(38,48)(40,59)(41,50)(47,52)(51,61)(53,64)(54,62)(56,63);(8,31)(13,41)(32,57)(33,39)(36,48)(42,61)(58,62)(59,60);(1,4,7,21)(2,9,11,34)(3,14,16,44)(5,24,27,56)(6,30,25,20)(8,32,31,57)(10,37,38,54)(12,40,28,35)(13,42,41,61)(15,46,47,64)(17,49,29,45)(18,50,51,22)(19,52,53,26)(23,43,55,63)(33,60,39,59)(36,58,48,62)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307114, #72307122, #72307560, …
maximal normal subgroup of	#72307114, #72307122, #72307560, …
maximal quotient of	C₂×#499744

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406940);