#184575

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×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

647 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)(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,23,24);(2,5)(3,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,26,28,31,29,32,36,39)(27,33,35,40,34,30,37,38);(1,3,4,7)(2,6,5,8)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,37)(39,40);(2,5)(6,8)(10,15)(11,17)(16,22)(18,23)(27,34)(30,33)(35,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91771);