#422126

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1029 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)(5,6,11,15)(7,17,20,29)(8,21,24,13)(9,26,19,12)(10,16,27,14)(18,33,35,30)(22,28,34,36)(23,31,25,32);(1,3,4,2)(5,7,11,20)(6,12,15,26)(8,22,24,34)(9,27,19,10)(13,30,21,33)(14,29,16,17)(18,23,35,25)(28,32,36,31);(2,3)(5,8,11,24)(6,13,15,21)(7,18,20,35)(9,22,19,34)(10,23,27,25)(12,28,26,36)(14,31,16,32)(17,30,29,33);(5,9,10,7,11,19,27,20)(6,12,14,17,15,26,16,29)(8,22,23,18,24,34,25,35)(13,28,31,30,21,36,32,33)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,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, 329322);