#152298

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59494);