#434241

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1183 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,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, 341437);