#6381862

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2664 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289058);