#508806

order	512
nilpotency class	4
isoclinism	isoclinic class of #508805
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30137
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 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)(3,14)(4,21,23,37)(5,26)(6,12)(7,11)(8,13)(9,17)(10,24,36,27)(15,42)(16,35,22,33)(18,40)(19,34)(20,39)(25,52,55,62)(28,44)(29,31)(30,46)(32,41)(38,56,61,58)(43,47,60,49)(45,48,53,54)(50,51)(57,64,59,63);(1,2)(3,15)(4,10)(5,11)(6,29)(7,26)(8,13)(9,18)(12,31)(14,42)(16,45)(17,40)(19,51)(20,46)(21,27)(22,53)(23,36)(24,37)(25,52)(28,41)(30,39)(32,44)(33,48)(34,50)(35,54)(38,56)(43,57)(47,63)(49,64)(55,62)(58,61)(59,60);(1,3,8,20)(2,9,13,34)(4,22,23,16)(5,19,7,17)(6,30,32,42)(10,35,36,33)(11,39,26,14)(12,40,41,51)(15,44,46,29)(18,28,50,31)(21,43,37,60)(24,47,27,49)(25,57,55,59)(38,63,61,64)(45,52,53,62)(48,56,54,58);(1,4,5,27,8,23,7,24)(2,10,11,21,13,36,26,37)(3,16,17,49,20,22,19,47)(6,25,28,58,32,55,31,56)(9,33,14,43,34,35,39,60)(12,38,29,52,41,61,44,62)(15,45,40,64,46,53,51,63)(18,48,42,57,50,54,30,59)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416002);