#420843

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
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

1913 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)(5,6,10,15)(7,16,20,30)(8,21,24,28)(9,13,25,14)(11,18,27,19)(12,22,29,23)(17,26,34,36)(31,33,35,32);(5,7,9,19,10,20,25,18)(6,11,14,16,15,27,13,30)(8,17,23,32,24,34,22,33)(12,26,21,35,29,36,28,31);(5,8)(6,12)(7,17)(9,22)(10,24)(11,26)(13,21)(14,28)(15,29)(16,31)(18,32)(19,33)(20,34)(23,25)(27,36)(30,35);(2,4)(6,13)(7,18)(9,25)(11,27)(12,21)(14,15)(17,32)(19,20)(22,23)(26,36)(28,29)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 328039);