#506545

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,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, 413741);