#401064

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1738 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=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, 308260);