#141068

order	512
nilpotency class	4
isoclinism	isoclinic class of #125269
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48264);