#129725

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 852 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)(5,6,10,17,12,16,11,34)(7,18,21,42,23,46,22,54)(8,24,26,41,28,14,27,40)(9,29,31,15,33,44,32,43)(13,30,37,53,36,52,38,20)(19,47,49,65,51,45,50,66)(25,57,59,39,61,62,60,56)(35,58,63,48,55,67,64,68);(5,7,12,23)(6,13,16,36)(8,19,28,51)(9,30,33,52)(10,22,11,21)(14,35,24,55)(15,42,43,54)(17,37,34,38)(18,44,46,29)(20,32,53,31)(25,58,61,67)(26,50,27,49)(39,65,56,66)(40,63,41,64)(45,57,47,62)(48,59,68,60);(2,4)(5,8)(6,14)(7,19)(9,25)(10,27)(11,26)(12,28)(13,35)(15,39)(16,24)(17,41)(18,45)(20,48)(21,50)(22,49)(23,51)(29,62)(30,58)(31,60)(32,59)(33,61)(34,40)(36,55)(37,64)(38,63)(42,65)(43,56)(44,57)(46,47)(52,67)(53,68)(54,66)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36921);