#481067

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#578	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322712, #72322724, #72322732, …
maximal normal subgroup of	#72322712, #72322724, #72322732, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388263);