#477837

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

716 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)(5,6,11,7,10,15,12,16)(8,19,21,29,23,17,22,13)(9,18,26,14,25,30,27,24)(20,33,34,32,36,28,35,31);(5,7,11,15,10,16,12,6)(8,19,21,29,23,17,22,13)(9,24,26,18,25,14,27,30)(20,28,34,31,36,33,35,32);(5,8,12,21,10,23,11,22)(6,13,16,19,15,29,7,17)(9,20,27,34,25,36,26,35)(14,28,18,31,24,33,30,32);(2,4)(5,9,10,25)(6,14,15,24)(7,18,16,30)(8,20,23,36)(11,27,12,26)(13,28,29,33)(17,31,19,32)(21,35,22,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385033);