#412283

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1262 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)(7,17,21,28)(8,19,23,20)(9,24)(10,14)(11,15)(12,16)(13,25)(18,32,35,22)(26,30)(27,29)(31,33,36,34);(1,2)(3,4)(5,7)(6,8)(9,18)(10,20)(11,19)(12,21)(13,22)(14,28)(15,17)(16,23)(24,31)(25,34)(26,33)(27,35)(29,36)(30,32);(2,4)(5,8)(6,7)(9,22)(10,17)(11,28)(12,23)(13,18)(14,19)(15,20)(16,21)(24,34)(25,31)(26,36)(27,32)(29,33)(30,35);(5,9,10,25,12,27,11,26)(6,13,15,24,16,30,14,29)(7,18,20,34,21,35,19,33)(8,22,17,31,23,32,28,36)

Polycyclic presentation:

2^9|a^2=ef,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,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, 319479);