#129115

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
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

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

normal subgroups only

This group has 238 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,g^b=gi,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, 36311);