#419182

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1910 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,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,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi

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, 326378);