#198084

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1404 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105280);