#492907

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2299 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=be,e^a=eh,g^a=gi,h^a=hi,f^b=fhi,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400103);