#494739

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,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, 401935);