#183752

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂³
commutator, socle	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 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)(11,15,20,23,19,18,22,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,36)(35,38)(39,40);(2,5)(6,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24)(25,27,28,33,29,34,35,39)(26,30,32,37,31,36,38,40);(9,12,13,21,14,10,16,17)(11,18,20,24,19,15,22,23);(1,3,4,7)(2,6,5,8)(11,19)(15,18)(20,22)(23,24)(26,31)(30,36)(32,38)(37,40)

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,d^a=di,d^b=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90948);