#217869

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,19)(13,18)(14,17)(15,22)(16,21)(23,24)(25,26,30,33,31,28,29,34)(27,32,36,39,37,38,35,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,22)(14,21)(16,19)(17,24)(18,20)(25,27,30,36,31,37,29,35)(26,32,33,39,28,38,34,40);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24)(25,28)(26,31)(27,32)(29,33)(30,34)(35,40)(36,39)(37,38);(2,6)(5,7)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)(25,29,31,30)(26,33,28,34)(27,35,37,36)(32,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,d^b=dhi,d^c=di,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125065);