#506544

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	#25799	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 712 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:

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

Polycyclic presentation:

2^9|a^2=fi,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, 413740);