#482673

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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×C₄⋊C₄	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

879 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,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, 389869);