#422682

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,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, 329878);