#123673

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1478 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)(7,17,21,30)(8,22)(9,14)(10,15)(11,16)(12,19,27,20)(13,24)(18,26,34,28)(23,29)(25,31)(32,35,36,33);(1,2)(3,4)(5,7,10,20,11,21,9,19)(6,12,14,17,16,27,15,30)(8,18,24,35,25,34,23,33)(13,28,22,36,29,26,31,32);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,22,16,29,15,31)(7,18,19,33,21,34,20,35)(12,26,17,32,27,28,30,36)

Polycyclic presentation:

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

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, 30869);