#396122

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

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

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, 303318);