#412087

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1380 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)(6,12)(7,16)(8,21)(9,10)(13,17)(14,28)(15,24)(18,32)(19,20)(22,23)(27,36)(31,35);(5,6)(7,17)(8,14)(9,24)(10,12)(11,15)(13,20)(16,29)(18,31)(19,25)(21,26)(22,34)(23,28)(27,32)(30,36)(33,35);(2,4)(5,7,9,19,11,20,10,16)(6,13,12,25,15,17,24,29)(8,18,22,33,23,32,21,30)(14,27,26,35,28,31,34,36);(5,8,9,22,11,23,10,21)(6,14,24,34,15,28,12,26)(7,18,19,33,20,32,16,30)(13,27,29,36,17,31,25,35)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,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, 319283);