#483594

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320804, #72320847, #72320855, …
maximal normal subgroup of	#72320804, #72320847, #72320855, …
maximal quotient of	#72343991

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390790);