#420379

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1293 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^c=hi,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, 327575);