#481605

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1133 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,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, 388801);