#146860

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

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

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

normal subgroups only

This group has 684 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)(7,18,20,34)(8,22,21,13)(9,15)(10,16,25,32)(11,27)(12,17)(14,23)(19,33)(24,31,26,28)(29,35)(30,36);(1,2,3,4)(5,7,11,21,12,20,9,8)(6,13,15,18,17,22,27,34)(10,19,26,35,25,36,24,23)(14,28,30,16,29,31,33,32);(5,8,9,20,12,21,11,7)(6,14,15,30,17,29,27,33)(10,23,24,36,25,35,26,19)(13,28,18,16,22,31,34,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54056);