#427839

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,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

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, 335035);