#492901

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×D₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2323 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)(7,12)(8,13)(9,14,22,11)(10,15)(16,25)(17,26,32,23)(18,27)(19,28)(20,29,34,24)(21,30)(31,36,33,35);(1,2,3,4)(6,11,15,14)(12,23,27,26)(13,24,30,29)(25,35,28,36);(5,7,10,18)(6,12,15,27)(8,19,21,16)(9,17,22,32)(11,23,14,26)(13,28,30,25)(20,33,34,31)(24,36,29,35);(5,8,10,21)(6,13,15,30)(7,16,18,19)(9,20,22,34)(11,24,14,29)(12,25,27,28)(17,31,32,33)(23,35,26,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^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, 400097);