#427982

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1234 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=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, 335178);