#493360

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3374 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)(7,15)(8,12)(9,22,23,13)(10,14)(11,17,25,32)(16,29)(18,30)(19,33)(20,35,36,27)(21,28)(24,31,26,34);(5,7,10,18)(6,11,14,25)(8,19,21,16)(9,17,23,32)(12,26,28,24)(13,15,22,30)(20,34,36,31)(27,33,35,29);(2,4);(5,8,10,21)(6,12,14,28)(7,16,18,19)(9,20,23,36)(11,24,25,26)(13,27,22,35)(15,29,30,33)(17,31,32,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,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, 400556);