#434235

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341431);