#205551

order	512
nilpotency class	3
isoclinism	isoclinic class of #205546
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60673	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

992 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112747);