#379814

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 287010);