#130047

order	512
nilpotency class	4
isoclinism	isoclinic class of #124718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

572 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 102 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)(5,6,10,15,9,14,28,41,11,17,32,45,29,16,46,44)(7,18,21,48,20,36,53,66,22,35,55,42,27,12,30,34)(8,23,26,57,25,40,60,43,24,38,31,63,58,13,37,39)(19,49,52,65,51,56,62,67,50,33,54,64,59,47,61,68);(5,7,11,22)(6,12,17,36)(8,19,24,50)(9,27,29,20)(10,30,32,53)(13,33,40,49)(14,35,16,18)(15,42,45,48)(21,28,55,46)(23,56,38,47)(25,59,58,51)(26,61,31,62)(34,41,66,44)(37,52,60,54)(39,67,43,68)(57,65,63,64);(2,4)(5,8)(6,13,17,40)(7,19)(9,25)(10,31)(11,24)(12,33,36,49)(14,23,16,38)(15,43,45,39)(18,47,35,56)(20,51)(21,54)(22,50)(26,32)(27,59)(28,37)(29,58)(30,62)(34,65,66,64)(41,63,44,57)(42,68,48,67)(46,60)(52,55)(53,61)

Polycyclic presentation:

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

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, 37243);