#146871

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
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	#1820
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 686 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)(7,17,10,15)(8,20)(9,14)(11,16)(12,19,28,24)(13,18,31,21)(22,29)(23,32)(25,27)(26,36)(30,33,34,35);(1,2,3,4)(5,7,11,10)(6,12)(8,21,23,18)(9,19,25,24)(13,29)(14,17)(15,27)(16,28)(20,34)(22,35,36,33)(26,31)(30,32);(5,8,9,22,11,23,25,36)(6,13,14,30,16,31,27,34)(7,18,19,33,10,21,24,35)(12,26,17,20,28,29,15,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fi,e^c=ei,f^c=fhi,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, 54067);