#482373

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	#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

1062 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)(11,19)(12,16)(13,27,28,17)(14,18)(15,22,31,21)(20,35)(23,30)(24,38)(25,39,40,33)(26,34)(29,37,32,36);(2,5)(3,8)(6,7)(9,11)(10,15)(12,24)(13,22)(14,23)(16,32)(17,19)(18,31)(20,26)(21,28)(25,36)(27,30)(29,34)(33,38)(35,39)(37,40);(2,6)(5,7)(9,12,14,26)(10,16,18,34)(11,20,23,24)(13,25,28,40)(15,29,31,32)(17,33,27,39)(19,35,30,38)(21,36,22,37);(2,6)(5,7)(9,13)(10,17)(11,21)(12,25)(14,28)(15,30)(16,33)(18,27)(19,31)(20,36)(22,23)(24,37)(26,40)(29,38)(32,35)(34,39)

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=h,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.

Relationships

maximal subgroup of	#72284971, #72285068
maximal normal subgroup of	#72284971, #72285068

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389569);