#510079

order	512
nilpotency class	4
isoclinism	isoclinic class of #510079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31260
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24287	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417275);