#152637

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	C₂×#2215	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

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

normal subgroups only

This group has 341 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,9,15,12,17,11,31)(7,18,20,26,14,33,29,35)(8,22,25,36,28,21,27,32)(10,19,24,34,30,23,13,16);(5,7,18,31,10,21,8,23,9,29,26,17,24,36,25,34,12,14,33,15,30,22,28,19,11,20,35,6,13,32,27,16);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,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.

Relationships

maximal subgroup of	#72319455, #72319630, D₄×#1419, …
maximal normal subgroup of	#72319455, #72319630, D₄×#1419

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59833);