#146863

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1799
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fi,f^c=fh,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, 54059);