#494734

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401930);