#152200

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 310 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)(5,6,8,14,9,15,22,34,10,16,23,35,24,36,45,56,11,13,21,31,25,32,43,53,26,33,44,54,46,55,61,66)(7,12,18,28,19,29,40,50,20,30,41,51,42,52,60,65,17,27,37,47,38,48,57,62,39,49,58,63,59,64,67,68);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,27)(14,28)(15,29)(16,30)(21,37)(22,40)(23,41)(24,42)(25,38)(26,39)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(43,57)(44,58)(45,60)(46,59)(53,62)(54,63)(55,64)(56,65)(61,67)(66,68);(6,13)(7,17)(8,21)(15,32)(16,33)(19,38)(20,39)(22,43)(23,44)(28,47)(36,55)(42,59)(45,61)(50,62)(51,63)(65,68)

Polycyclic presentation:

2^9|a^2=f,b^2=d,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ci,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59396);