#123994

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1442
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1546 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)(5,6,9,16,11,14,10,15)(7,17,20,32,21,12,19,28)(8,22,25,13,24,30,26,31)(18,29,23,27,34,36,35,33);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,28,14,17,16,32)(8,23,25,18,24,35,26,34)(13,29,30,33,31,36,22,27);(5,8,9,25,11,24,10,26)(6,13,15,22,14,31,16,30)(7,18,19,35,21,34,20,23)(12,27,28,36,17,33,32,29)

Polycyclic presentation:

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

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, 31190);