#441216

order	512
nilpotency class	3
isoclinism	isoclinic class of #441216
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15674
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348412);