#138169

order	512
nilpotency class	4
isoclinism	isoclinic class of #125269
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

471 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 64

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

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

Polycyclic presentation:

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

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, 45365);