#135736

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1568
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 130 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)(5,6,11,18,10,17,35,45,13,19,34,16,12,38,9,33)(7,20,24,40,23,49,22,41,26,46,36,62,25,58,56,14)(8,27,30,60,29,44,28,47,32,61,37,63,31,15,42,43)(21,50,53,39,52,64,66,59,55,68,57,65,54,48,51,67);(5,7,9,22,12,25,34,24,13,26,35,56,10,23,11,36)(6,14,16,20,17,40,33,49,19,41,18,46,38,62,45,58)(8,21,28,51,31,54,30,57,32,55,42,66,29,52,37,53)(15,39,43,48,27,59,60,50,44,65,47,64,61,67,63,68);(1,2,3,4)(5,8)(6,15,19,44)(7,21)(9,28)(10,29)(11,37)(12,31)(13,32)(14,39,41,65)(16,43,18,47)(17,27,38,61)(20,48,46,64)(22,51)(23,52)(24,57)(25,54)(26,55)(30,34)(33,60,45,63)(35,42)(36,53)(40,59,62,67)(49,50,58,68)(56,66)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42932);