#465312

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

777 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)(5,6,11,14)(7,15,18,33)(8,13,23,21)(9,10,25,20)(12,27,29,17)(16,34,19,32)(22,24,26,31)(28,36,30,35);(5,7,10,17,11,18,20,27)(6,12,9,15,14,29,25,33)(8,19,26,35,23,16,22,36)(13,30,24,34,21,28,31,32);(5,8,20,22,11,23,10,26)(6,13,25,31,14,21,9,24)(7,16,17,36,18,19,27,35)(12,28,15,32,29,30,33,34);(2,4)(5,9,10,14,11,25,20,6)(7,15,17,29,18,33,27,12)(8,21,22,24,23,13,26,31)(16,30,35,32,19,28,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372508);