#407055

order	512
nilpotency class	3
isoclinism	isoclinic class of #407055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14946
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1127 conjugacy classes of subgroups — too many to draw. Only its 224 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)(3,8)(4,19)(5,10,24,36)(6,27)(7,12)(9,22)(11,13)(14,40)(15,44,45,29)(16,31)(17,30)(18,32)(20,48)(21,52,53,49)(23,50)(25,38,26,37)(28,42)(33,43)(34,55,60,54)(35,39)(41,61,62,59)(46,57,47,58)(51,64,63,56);(1,3,13,16,7,18,6,17)(2,8,11,31,12,32,27,30)(4,20,22,43,23,42,39,14)(5,15,25,47,24,45,26,46)(9,33,50,28,35,40,19,48)(10,29,38,57,36,44,37,58)(21,51,55,62,53,63,54,41)(34,59,52,56,60,61,49,64);(1,4,6,22,7,23,13,39)(2,9,11,19,12,35,27,50)(3,14,16,42,18,43,17,20)(5,21,26,55,24,53,25,54)(8,28,30,40,32,48,31,33)(10,34,38,49,36,60,37,52)(15,41,47,63,45,62,46,51)(29,56,58,61,44,64,57,59);(1,5)(2,10)(3,15)(4,21)(6,26)(7,24)(8,29)(9,34)(11,38)(12,36)(13,25)(14,41)(16,47)(17,46)(18,45)(19,49)(20,51)(22,55)(23,53)(27,37)(28,56)(30,58)(31,57)(32,44)(33,59)(35,60)(39,54)(40,61)(42,63)(43,62)(48,64)(50,52)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314251);