#407028

order	512
nilpotency class	3
isoclinism	isoclinic class of #407027
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

1066 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 1066 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)(3,15)(4,20)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,28)(16,30)(17,41)(19,39)(21,49)(22,36)(24,35)(26,37)(27,38)(29,32)(31,48)(33,44)(34,50)(40,43)(42,58)(45,53)(46,57)(47,56)(51,54)(52,59)(55,62)(60,61)(63,64);(1,3,8,19)(2,9,14,32)(4,16)(5,17,28,46)(6,29,7,18)(10,33)(11,31,25,53)(12,39,13,15)(20,40)(21,42)(22,47)(23,43)(24,44)(26,48,27,45)(30,36)(34,58)(35,56)(37,41,38,57)(49,60)(50,63)(51,61)(52,62)(54,55)(59,64);(1,4)(2,10)(3,16,19,44)(5,21)(6,23)(7,22)(8,24)(9,30,32,33)(11,34)(12,20)(13,35)(14,36)(15,40,39,56)(17,42,46,62)(18,47,29,43)(25,54)(26,51)(27,50)(28,52)(31,55,53,58)(37,59)(38,49)(41,60,57,64)(45,63,48,61);(1,5,8,28)(2,11,14,25)(3,17,19,46)(4,21,24,52)(6,27,7,26)(9,31,32,53)(10,34,36,54)(12,38,13,37)(15,41,39,57)(16,42,44,62)(18,48,29,45)(20,49,35,59)(22,51,23,50)(30,55,33,58)(40,60,56,64)(43,63,47,61)

Polycyclic presentation:

2^9|b^2=efh,c^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 314224);