#140893

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 451 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=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, 48089);