#481065

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂²×#135	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216710, #72216933, #72216980, …
maximal normal subgroup of	#72216710, #72216933, #72216980, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388261);