#140261

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

211 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=d,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^c=eghi,f^c=fi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47457);