#481682

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1106 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)(13,17)(14,18)(15,20,29,36)(21,33)(22,34)(24,37,39,30)(26,40)(27,32)(28,35,31,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,18,29)(12,20,25,36)(13,26,27,21)(16,19,23,34)(17,31,32,28)(24,38,39,35)(30,40,37,33);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,19)(17,30)(18,23)(21,35)(22,36)(26,38)(27,39)(28,33)(29,34)(31,40)(32,37);(9,13,14,27)(10,17,18,32)(11,21,22,26)(12,24,25,39)(15,28,29,31)(16,30,23,37)(19,33,34,40)(20,35,36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,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, 388878);