#429472

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,f^c=fi,g^c=gi,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, 336668);