#412081

order	512
nilpotency class	3
isoclinism	isoclinic class of #396373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1358 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^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,e^b=ei,f^b=fi,g^b=gi,f^c=fi

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, 319277);