#142809

order	512
nilpotency class	4
isoclinism	isoclinic class of #142258
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

622 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,12,19)(7,20,25,54)(8,14,30,46)(9,31,36,49)(10,18,11,17)(13,23,40,22)(15,35,16,34)(21,53,56,68)(24,50,37,47)(26,61,62,67)(27,51,64,48)(28,45,29,44)(32,52,59,39)(33,43,63,42)(38,58,66,57)(41,55,65,60);(5,7)(6,13)(8,26)(9,32)(10,37)(11,24)(12,25)(14,41)(15,47)(16,50)(17,52)(18,39)(19,40)(20,31)(21,27)(22,34)(23,35)(28,58)(29,57)(30,62)(33,55)(36,59)(38,43)(42,66)(44,53)(45,68)(46,65)(48,67)(49,54)(51,61)(56,64)(60,63);(2,4)(5,8,12,30)(6,14,19,46)(7,21,25,56)(9,33,36,63)(10,29,11,28)(13,38,40,66)(15,48,16,51)(17,45,18,44)(20,53,54,68)(22,57,23,58)(24,60,37,55)(26,59,62,32)(27,34,64,35)(31,43,49,42)(39,67,52,61)(41,47,65,50)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50005);