#152164

order	512
nilpotency class	5
isoclinism	isoclinic class of #150138
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 106 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)(5,6,8,14,12,17,11,35)(7,18,21,51,23,53,19,49)(9,26,24,61,30,48,29,66)(10,31,25,47,34,44,33,16)(13,36,38,20,40,56,37,58)(15,42,41,54,45,65,63,28)(22,59,55,43,60,68,32,52)(27,46,64,67,57,50,62,39);(5,7)(6,13)(8,19)(9,27)(10,32)(11,21)(12,23)(14,37)(15,43)(16,46)(17,40)(18,48)(20,54)(22,33)(24,62)(25,60)(26,53)(28,58)(29,64)(30,57)(31,39)(34,55)(35,38)(36,65)(41,59)(42,56)(44,67)(45,52)(47,50)(49,66)(51,61)(63,68);(2,4)(5,8,12,11)(6,14,17,35)(7,19,23,21)(9,24,30,29)(10,25,34,33)(13,37,40,38)(15,41,45,63)(16,31,47,44)(18,49,53,51)(20,36,58,56)(22,32,60,55)(26,61,48,66)(27,62,57,64)(28,42,54,65)(39,50,67,46)(43,59,52,68)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fi,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59360);