#422202

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1052 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 329398);