#482119

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,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, 389315);