#506543

order	512
nilpotency class	4
isoclinism	isoclinic class of #506543
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33531	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^c=fh,g^c=gh,h^c=hi,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, 413739);