#402447

order	512
nilpotency class	3
isoclinism	isoclinic class of #402446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1005 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)(5,6,11,16,12,17,10,19)(7,18,28,8,21,23,14,13)(9,24,26,31,27,33,25,15)(20,22,35,36,34,29,30,32);(5,7,10,14,12,21,11,28)(6,13,16,18,17,8,19,23)(9,20,25,30,27,34,26,35)(15,29,24,32,31,22,33,36);(5,8,12,13)(6,14)(7,19)(9,22,27,29)(10,23,11,18)(15,30)(16,21)(17,28)(20,33)(24,34)(25,36,26,32)(31,35);(2,4)(5,9,10,25,12,27,11,26)(6,15,16,24,17,31,19,33)(7,20,14,30,21,34,28,35)(8,22,23,36,13,29,18,32)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,d^2=e,e^2=i,f^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,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309643);