#511969

order	512
nilpotency class	4
isoclinism	isoclinic class of #511969
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32930
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|b^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419165);