#396373

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	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1415 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)(5,6,10,14,11,16,9,15)(7,17,21,28,22,18,20,12)(8,23,25,13,26,29,24,30)(19,27,34,31,35,36,33,32);(5,7,9,20,11,22,10,21)(6,12,14,17,16,28,15,18)(8,19,24,33,26,35,25,34)(13,27,29,31,30,36,23,32);(5,6)(7,18)(8,13)(9,14)(10,15)(11,16)(12,20)(17,22)(19,32)(21,28)(23,25)(24,29)(26,30)(27,33)(31,35)(34,36);(2,4)(5,8,9,24,11,26,10,25)(6,13,14,29,16,30,15,23)(7,19,20,33,22,35,21,34)(12,27,17,31,28,36,18,32)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=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, 303569);