#6409733

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	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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)(3,7)(4,5)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26,31,35,30,32,29,38)(27,33,36,40,28,34,37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,32)(33,34)(35,38)(39,40);(2,5)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,30,28)(26,33,32,34)(29,37,31,36)(35,40,38,39);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28,30,27)(26,34,32,33)(29,36,31,37)(35,39,38,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,30,31)(26,35,32,38)(27,36,28,37)(33,39,34,40)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^c=ei,e^d=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, 6316929);