#420385

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1351 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)(3,4)(5,6,9,14,10,15,23,11)(7,16,20,27,17,26,19,12)(8,13,21,28,22,29,34,24)(18,30,32,36,31,35,33,25);(6,11)(7,17)(8,21)(9,23)(12,16)(14,15)(18,32)(22,34)(24,28)(26,27)(30,35)(31,33);(5,7,9,19,10,17,23,20)(6,12,14,26,15,27,11,16)(8,18,21,33,22,31,34,32)(13,25,28,35,29,36,24,30);(2,4)(5,8,9,21,10,22,23,34)(6,13,14,28,15,29,11,24)(7,18,19,33,17,31,20,32)(12,25,26,35,27,36,16,30)

Polycyclic presentation:

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

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, 327581);