#152636

order	512
nilpotency class	5
isoclinism	isoclinic class of #150157
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	SD₃₂
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

315 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 96 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)(5,6,9,15,12,17,11,35)(7,18,21,48,23,42,31,50)(8,24,27,37,30,62,29,64)(10,32,26,46,34,43,33,16)(13,36,38,20,41,55,40,25)(14,19,44,52,45,53,61,28)(22,58,57,49,59,67,54,60)(39,56,47,51,68,63,65,66);(5,7,19,51,10,22,30,20,9,31,52,56,26,54,29,36,12,23,53,66,34,59,8,25,11,21,28,63,33,57,27,55)(6,13,37,67,16,39,14,42,35,38,24,60,43,47,61,50,17,41,64,58,46,68,45,18,15,40,62,49,32,65,44,48);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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=di,f^b=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, 59832);