#427946

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1251 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)(7,17,20,21)(8,22,24,25)(12,26,18,28)(13,29,23,30)(19,33)(27,32)(31,34)(35,36);(5,6,10,15)(7,18,20,12)(8,23,24,13)(9,16,11,14)(17,26,21,28)(19,32,33,27)(22,29,25,30)(31,35,34,36);(5,7)(6,12)(8,19)(9,21)(10,20)(11,17)(13,27)(14,28)(15,18)(16,26)(22,34)(23,32)(24,33)(25,31)(29,36)(30,35);(2,4)(5,8,9,22,10,24,11,25)(6,13,14,29,15,23,16,30)(7,19,21,34,20,33,17,31)(12,27,28,36,18,32,26,35)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 335142);