#422139

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

1042 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)(5,6,10,14,11,16,9,15)(7,17,20,12,21,29,22,18)(8,23,26,13,27,31,25,32)(19,28,24,36,34,33,35,30);(5,7,10,20,11,21,9,22)(6,12,14,29,16,18,15,17)(8,24,26,34,27,35,25,19)(13,30,31,28,32,36,23,33);(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,21)(17,22)(19,33)(20,29)(23,26)(24,36)(25,31)(27,32)(28,34)(30,35);(2,4)(5,8,9,25,11,27,10,26)(6,13,14,31,16,32,15,23)(7,19,20,24,21,34,22,35)(12,28,17,30,18,33,29,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 329335);