#424594

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1288 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)(5,6,10,13)(7,16)(8,14,23,28)(9,15,11,25)(12,19)(17,27)(18,29)(20,32)(21,36)(22,30,24,31)(26,34)(33,35);(5,7)(6,12)(8,21)(9,19)(10,20)(11,17)(13,27)(14,29)(15,32)(16,25)(18,22)(23,34)(24,35)(26,30)(28,33)(31,36);(6,13)(7,17,20,19)(12,16,27,32)(14,28)(15,25)(18,34,35,21)(26,33,36,29)(30,31);(2,4)(5,8)(6,14)(7,18)(9,24)(10,23)(11,22)(12,26)(13,28)(15,31)(16,33)(17,34)(19,21)(20,35)(25,30)(27,36)(29,32)

Polycyclic presentation:

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

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, 331790);