#6374824

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,33,28,32,36,40)(27,30,35,37,34,31,38,39);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,28,34)(26,30,32,31)(29,35,36,38)(33,37,40,39);(2,6)(5,8)(11,18)(14,17)(19,24)(21,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,38)(37,40);(10,15)(14,17)(16,22)(21,23)(27,34)(30,31)(35,38)(37,39);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)(27,34)(30,31)(35,38)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282020);