#149112

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#387
commutator	C₂²×C₄	C₂²×C₈
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

434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,g^b=gi,e^c=ei,f^c=fi,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, 56308);