#183803

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,12,16,21)(11,15,19,22,18,20,23,24)(25,26,29,33,30,34,37,40)(27,31,28,32,35,38,36,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,18)(16,23)(17,22)(21,24)(25,27,29,28,30,35,37,36)(26,31,33,32,34,38,40,39);(9,12,13,21,14,10,16,17)(11,15,19,22,18,20,23,24)(25,28,29,35,30,36,37,27)(26,32,33,38,34,39,40,31);(1,3,4,7)(2,6,5,8)(11,18)(15,20)(19,23)(22,24)(27,35)(28,36)(31,38)(32,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^b=ch,d^b=dh,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90999);