#379813

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	#35676	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2138 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:

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

Polycyclic presentation:

2^9|a^2=h,b^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, 287009);