#422140

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,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, 329336);