#6342466

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 524288
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

2176 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:

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

Polycyclic presentation:

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

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, 6249662);