#138170

order	512
nilpotency class	4
isoclinism	isoclinic class of #125269
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

421 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 108 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)(5,6,10,16,11,15,31,33,9,18,19,46,12,17,32,45)(7,13,22,36,21,37,47,65,24,35,54,59,23,38,29,43)(8,14,26,40,27,42,56,67,25,41,30,44,28,39,57,60)(20,48,50,64,52,63,58,66,51,34,53,61,49,62,55,68);(5,7,19,47,11,23,32,22,9,24,10,29,12,21,31,54)(6,13,33,59,17,37,16,43,18,35,45,36,15,38,46,65)(8,20,26,55,27,49,56,53,25,51,30,58,28,52,57,50)(14,34,60,61,39,62,44,68,41,48,67,64,42,63,40,66);(2,4)(5,8)(6,14)(7,20)(9,25)(10,30)(11,27)(12,28)(13,34)(15,42)(16,44)(17,39)(18,41)(19,26)(21,52)(22,53)(23,49)(24,51)(29,58)(31,57)(32,56)(33,60)(35,48)(36,64)(37,62)(38,63)(40,46)(43,68)(45,67)(47,55)(50,54)(59,61)(65,66)

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=gh,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fh,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, 45366);