#477321

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17109
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

859 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 859 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,f^d=fi,h^d=hi

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, 384517);