#482686

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#36032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

907 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,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, 389882);