#6342518

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁵
commutator, socle	C₂²	C₂³×C₄²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,c^a=ch,c^b=chi,e^d=eh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249714);