#129722

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

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

normal subgroups only

This group has 860 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)(7,18,21,45)(8,24,26,14)(9,29,31,43)(10,17)(11,34)(12,16)(13,20)(15,32,41,33)(19,46)(22,40,23,53)(25,57)(27,56,28,39)(30,42)(35,47,54,67)(36,52)(37,51)(38,59)(44,50)(48,66)(49,64)(55,60)(58,63,68,65)(61,62);(5,7,12,21)(6,13,16,36)(8,19,26,50)(9,30,31,51)(10,23,11,22)(14,35,24,54)(15,40,41,53)(17,42,34,37)(18,43,45,29)(20,32,52,33)(25,58,61,68)(27,49,28,48)(38,64,55,66)(39,65,56,63)(44,57,46,62)(47,59,67,60);(2,4)(5,8)(6,14)(7,19)(9,25)(10,27)(11,28)(12,26)(13,35)(15,38)(16,24)(17,39)(18,44)(20,47)(21,50)(22,48)(23,49)(29,62)(30,58)(31,61)(32,59)(33,60)(34,56)(36,54)(37,63)(40,64)(41,55)(42,65)(43,57)(45,46)(51,68)(52,67)(53,66)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,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, 36918);