#397209

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^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^b=gi,f^c=fi,g^c=gi,h^d=hi

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, 304405);