#422714

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

1032 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)(5,6,12,16,11,17,10,30)(7,18,21,13,23,32,22,31)(8,24,26,29,27,14,15,9)(19,33,25,36,35,20,34,28);(5,7,12,21,11,23,10,22)(6,13,16,32,17,31,30,18)(8,25,26,35,27,34,15,19)(9,28,24,33,29,36,14,20);(5,8,12,15,11,27,10,26)(6,14,30,9,17,24,16,29)(7,19,22,25,23,35,21,34)(13,28,32,20,31,36,18,33);(2,4)(5,9,10,14,11,29,12,24)(6,15,16,8,17,26,30,27)(7,20,21,28,23,33,22,36)(13,25,18,19,31,34,32,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,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, 329910);