#147798

order	512
nilpotency class	4
isoclinism	isoclinic class of #13429
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1009 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1009 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,6,11,17,12,16,10,34)(7,18,22,50,23,44,21,55)(8,24,28,48,29,14,27,43)(9,30,33,46,32,15,35,47)(13,36,39,31,40,56,38,20)(19,51,53,66,54,49,25,58)(26,42,62,68,61,45,63,67)(37,59,57,64,65,60,41,52);(5,7,12,23)(6,13,16,40)(8,25,29,53)(9,31,32,20)(10,22,11,21)(14,41,24,57)(15,44,30,18)(17,39,34,38)(19,28,54,27)(26,60,61,59)(33,36,35,56)(37,48,65,43)(42,58,45,66)(46,50,47,55)(49,68,51,67)(52,62,64,63);(2,4)(5,8,12,29)(6,14,16,24)(7,19,23,54)(9,26,32,61)(10,28,11,27)(13,37,40,65)(15,45,30,42)(17,48,34,43)(18,49,44,51)(20,52,31,64)(21,53,22,25)(33,63,35,62)(36,60,56,59)(38,57,39,41)(46,67,47,68)(50,66,55,58)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54994);