#152165

order	512
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 106 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)(6,11)(7,15)(8,18)(9,21)(12,24)(13,27)(14,28)(16,20)(17,29)(19,34)(22,33)(23,25)(26,35)(30,31)(32,36);(5,6)(7,16)(8,20)(9,23)(10,12)(11,15)(13,18)(14,21)(17,32)(19,35)(22,36)(24,29)(25,34)(26,33)(27,30)(28,31);(7,17)(11,24)(15,29)(16,32)(19,30)(25,28)(27,35)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59361);