#152844

order	512
nilpotency class	5
isoclinism	isoclinic class of #151127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

226 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,6,11,18,14,20,13,7)(8,15,25,21,22,30,16,10)(9,27,29,26,32,23,31,17)(12,33,28,36,35,24,34,19);(5,7,21,23,12,24,9,8,11,20,30,26,28,36,29,16,14,18,10,27,35,33,32,22,13,6,15,17,34,19,31,25);(2,4)(5,8,9,19,28,27,10,20,13,25,31,33,12,26,30,7,14,22,32,36,34,23,21,6,11,16,29,24,35,17,15,18)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=d,d^2=ghi,e^2=g,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319455, #72319593, #72319605
maximal normal subgroup of	#72319455, #72319593, #72319605

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60040);