#420775

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1836 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=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, 327971);