#6410957

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	#34876	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(3,8)(4,7)(5,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,29,33,30,34,35,40)(27,31,28,37,36,39,38,32);(2,6)(3,4)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27)(26,31)(28,29)(30,36)(32,40)(33,37)(34,39)(35,38);(1,4,5,3)(2,8,6,7)(11,20)(15,18)(19,21)(23,24)(25,28)(26,32)(27,35)(29,36)(30,38)(31,33)(34,37)(39,40);(2,6)(7,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,36)(28,38)(31,39)(32,37)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,d^c=di,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, 6318153);