#202486

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109682);