#152763

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 68 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)(5,6,10,15,9,17,27,44)(7,18,21,48,23,12,33,36)(8,13,25,39,24,40,30,43)(11,14,32,34,28,16,31,46)(19,35,50,61,51,47,53,63)(20,37,29,42,22,45,54,64)(26,41,58,67,55,38,57,60)(49,62,59,66,52,65,56,68);(5,7,10,29)(6,12,34,48)(8,19,30,59)(9,23,27,54)(11,22,32,21)(13,35,60,61)(14,37,44,64)(15,42,16,45)(17,18,46,36)(20,31,33,28)(24,51,25,56)(26,49,57,53)(38,65,43,68)(39,66,41,62)(40,47,67,63)(50,55,52,58);(2,4)(5,8,9,24)(6,13,17,40)(7,19,23,51)(10,30,27,25)(11,26,28,55)(12,35,18,47)(14,41,16,38)(15,43,44,39)(20,52,22,49)(21,53,33,50)(29,59,54,56)(31,58,32,57)(34,60,46,67)(36,63,48,61)(37,62,45,65)(42,68,64,66)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59959);