#494735

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	#35676	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

2042 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)(5,6,10,14,11,15,26,19)(7,16,13,25,20,9,24,17)(8,12,23,29,21,27,34,33)(18,30,28,36,32,22,35,31);(5,7,17,6,11,20,25,15)(8,18,31,12,21,32,36,27)(9,19,26,24,16,14,10,13)(22,33,34,35,30,29,23,28);(2,4)(5,8)(6,12)(7,18)(9,22)(10,23)(11,21)(13,28)(14,29)(15,27)(16,30)(17,31)(19,33)(20,32)(24,35)(25,36)(26,34);(8,21)(12,27)(18,32)(22,30)(23,34)(28,35)(29,33)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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, 401931);