#482401

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1089 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,26,27,17)(14,18)(15,21,29,38)(20,33)(22,34)(23,31,24,30)(25,32)(28,36,40,35)(37,39);(2,5)(3,8)(6,7)(9,11,14,22)(10,15,18,29)(12,20,25,37)(13,21,27,38)(16,28,32,40)(17,19,26,34)(23,35,24,36)(30,39,31,33);(2,6)(5,7)(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,31)(18,32)(19,33)(21,36)(22,37)(23,27)(26,30)(29,40)(34,39)(35,38);(2,6)(5,7)(9,13)(10,17)(11,21)(12,23)(14,27)(15,19)(16,30)(18,26)(20,35)(22,38)(24,25)(28,39)(29,34)(31,32)(33,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389597);