#511426

order	512
nilpotency class	4
isoclinism	isoclinic class of #509117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#32545
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,f^c=fi,g^c=gh,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, 418622);