#480254

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,f^2=h,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fg,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387450);