#134321

order	512
nilpotency class	4
isoclinism	isoclinic class of #11502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,e^c=ei,f^c=fg,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41517);