#379812

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2168 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,f^d=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, 287008);