#146859

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	#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

672 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^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.

Relationships

maximal subgroup of	#72313669, #72314009
maximal normal subgroup of	#72313669, #72314009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54055);