#492888

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,b^a=be,e^a=eh,h^a=hi,f^b=fh,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, 400084);