#494737

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2002 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

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, 401933);