#396360

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1407 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,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, 303556);