#6405308

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6312504);