#481793

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

985 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388989);