#146853

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1917
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,c^b=ch,d^b=di,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54049);