#441221

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1253 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)(5,6,9,16)(7,12)(8,21)(10,15,11,14)(13,25)(17,30,33,36)(18,29)(19,27)(20,28)(22,26,34,35)(23,31)(24,32);(5,7,10,20,9,18,11,19)(6,12,15,28,16,29,14,27)(8,22,25,17,23,34,24,33)(13,30,31,35,32,36,21,26);(2,4)(5,8)(6,13,16,32)(7,17)(9,23)(10,24)(11,25)(12,26,29,35)(14,31,15,21)(18,33)(19,34)(20,22)(27,30,28,36);(6,14,16,15)(7,18)(12,27,29,28)(13,31,32,21)(17,33)(19,20)(22,34)(26,30,35,36)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=ei,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=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, 348417);