#205660

order	512
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 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)(9,10,14,20,15,19,13,18)(11,16,22,31,23,33,29,32)(12,24,27,36,28,35,26,17)(21,37,38,30,25,39,40,34);(1,2,3,5,4,6,7,8)(9,11,14,29,15,23,13,22)(10,16,18,31,19,33,20,32)(12,25,27,38,28,21,26,40)(17,34,35,37,36,30,24,39);(2,6)(5,8)(9,12,13,26,15,28,14,27)(10,17,20,24,19,36,18,35)(11,21,22,38,23,25,29,40)(16,30,32,37,33,34,31,39);(9,13,15,14)(10,18,19,20)(11,22,23,29)(12,26,28,27)(16,31,33,32)(17,35,36,24)(21,38,25,40)(30,39,34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,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, 112856);