#482684

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	#36019	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

913 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)(11,19,22,32)(12,23,24,17)(13,26,28,18)(14,16)(15,21)(20,34)(25,31)(27,29)(30,36,37,40)(33,35)(38,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,16,29)(12,20,24,35)(13,27,28,21)(17,30,23,37)(18,32,26,19)(25,40,39,36)(31,34,38,33);(1,3,5,4)(2,6,7,8)(10,16)(12,24)(15,29)(18,26)(19,32)(20,35)(25,39)(36,40);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,20)(13,25)(14,24)(15,30)(16,23)(18,31)(19,33)(21,36)(22,35)(26,38)(27,40)(28,39)(29,37)(32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,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, 389880);