#123907

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1441
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237181, #72237183, #72237187, …
maximal normal subgroup of	#72237181, #72237183, #72237187, …
maximal quotient of	#72239100, #72239212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31103);