#152634

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	#33600	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 94 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)(5,6,9,15,12,17,11,34)(7,18,22,45,20,43,24,57)(8,25,27,36,30,59,29,60)(10,32,31,42,33,61,26,16)(13,35,38,50,37,54,62,21)(14,19,40,48,41,49,58,28)(23,56,55,65,51,68,52,46)(39,53,44,47,63,67,64,66);(5,7,19,47,33,51,8,21,9,22,48,67,26,52,27,35,12,20,49,66,10,23,30,50,11,24,28,53,31,55,29,54)(6,13,36,56,42,63,41,43,34,62,25,46,32,44,40,45,17,37,60,68,16,39,14,18,15,38,59,65,61,64,58,57);(2,4)(7,20)(13,37)(18,43)(21,50)(22,24)(23,51)(35,54)(38,62)(39,63)(44,64)(45,57)(46,65)(47,66)(52,55)(53,67)(56,68)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci

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, 59830);