#407012

order	512
nilpotency class	3
isoclinism	isoclinic class of #407011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14945
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

1079 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1079 conjugacy classes of subgroups — too many to draw. Only its 208 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)(3,8,15,27)(4,9,20,32)(5,10,21,33)(6,23,24,11)(13,35,37,25)(14,38,39,26)(16,42,41,28)(17,29,44,55)(18,45,46,30)(19,48,49,31)(22,51,50,34)(36,57,58,52)(40,53,60,59)(43,54,61,62)(47,56,64,63);(1,3)(2,8)(4,17,20,44)(5,14)(6,16)(7,15)(9,29,32,55)(10,26)(11,28)(12,27)(13,19,37,49)(18,43,46,61)(21,39)(22,40)(23,42)(24,41)(25,31,35,48)(30,54,45,62)(33,38)(34,53)(36,47,58,64)(50,60)(51,59)(52,56,57,63);(1,4)(2,9)(3,13,15,37)(5,18)(6,19)(7,20)(8,25,27,35)(10,30)(11,31)(12,32)(14,36,39,58)(16,17,41,44)(21,46)(22,47)(23,48)(24,49)(26,52,38,57)(28,29,42,55)(33,45)(34,56)(40,43,60,61)(50,64)(51,63)(53,54,59,62);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,26)(9,30)(11,34)(12,33)(13,36)(15,39)(16,40)(17,43)(19,47)(20,46)(23,51)(24,50)(25,52)(27,38)(28,53)(29,54)(31,56)(32,45)(35,57)(37,58)(41,60)(42,59)(44,61)(48,63)(49,64)(55,62)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,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, 314208);