#148683

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

506 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)(5,6,11,17,10,15,28,35,12,18,29,36,9,16,20,30)(7,19,22,34,23,33,8,25,24,31,26,14,21,13,27,32);(5,7,9,21,12,24,10,23)(6,13,15,19,18,33,16,31)(8,11,22,20,27,29,26,28)(14,17,32,35,34,36,25,30);(2,4)(5,8,9,22,12,27,10,26)(6,14,15,32,18,34,16,25)(7,20,21,29,24,28,23,11)(13,30,19,17,33,35,31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,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, 55879);