#427390

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
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

1231 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)(7,11)(8,20,22,12)(9,15)(10,14)(13,24,31,23)(16,27,35,32)(17,36,21,30)(18,29)(19,28)(25,34)(26,33);(1,2)(3,4)(5,7,10,18)(6,11,14,29)(8,21,22,17)(9,25,26,19)(12,30,20,36)(13,32,31,27)(15,34,33,28)(16,24,35,23);(2,4)(5,8)(6,12)(7,16)(9,24)(10,22)(11,27)(13,15)(14,20)(17,19)(18,35)(21,25)(23,26)(28,36)(29,32)(30,34)(31,33);(5,8)(6,13,14,31)(7,17,18,21)(9,24)(10,22)(11,27)(12,15,20,33)(16,19,35,25)(23,26)(28,36)(29,32)(30,34)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,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, 334586);