#366268

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

849 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273464);