#142812

order	512
nilpotency class	4
isoclinism	isoclinic class of #142310
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
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)(5,6,12,18)(7,19,22,50)(8,14,30,49)(9,31,34,16)(10,35,33,15)(11,38,32,17)(13,36,41,21)(20,45,56,53)(23,44,60,43)(24,54,25,42)(26,40,59,65)(27,47,37,63)(28,46,62,51)(29,52,39,48)(55,66,61,67)(57,64,58,68);(5,7)(6,13)(8,26)(9,23)(10,36)(11,25)(12,22)(14,45)(15,42)(16,50)(17,44)(18,41)(19,31)(20,28)(21,33)(24,32)(27,57)(29,55)(30,59)(34,60)(35,54)(37,58)(38,43)(39,61)(40,47)(46,66)(48,64)(49,53)(51,67)(52,68)(56,62)(63,65);(2,4)(5,8,12,30)(6,14,18,49)(7,20,22,56)(9,28,34,62)(10,37,33,27)(11,39,32,29)(13,40,41,65)(15,47,35,63)(16,51,31,46)(17,52,38,48)(19,53,50,45)(21,59,36,26)(23,61,60,55)(24,58,25,57)(42,68,54,64)(43,67,44,66)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dh,f^c=fg,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, 50008);