#491122

order	512
nilpotency class	4
isoclinism	isoclinic class of #491120
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#19258
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33692	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|b^2=egi,d^2=g,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398318);