#422713

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1064 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)(5,6,12,8,11,16,10,15)(7,17,20,13,21,22,28,18)(9,23,27,32,26,31,25,14)(19,33,24,30,34,29,36,35);(5,7,12,20,11,21,10,28)(6,13,8,22,16,18,15,17)(9,24,27,34,26,36,25,19)(14,30,23,29,32,35,31,33);(5,8,10,16,11,15,12,6)(7,18,20,22,21,13,28,17)(9,23,25,32,26,31,27,14)(19,33,24,35,34,29,36,30);(2,4)(5,9,10,25,11,26,12,27)(6,14,8,23,16,32,15,31)(7,19,20,24,21,34,28,36)(13,29,17,30,18,33,22,35)

Polycyclic presentation:

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

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, 329909);