#197618

order	512
nilpotency class	3
isoclinism	isoclinic class of #19384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#3035	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1363 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:

(2,5)(3,6)(7,8)(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);(1,2,6,8,4,7,3,5)(9,11,14,20)(10,15)(12,19,23,36)(13,25,27,21)(16,28)(17,30)(18,33)(22,39)(24,38,26,37)(29,32)(31,40)(34,35);(9,12)(10,16)(11,19)(13,26)(14,23)(15,28)(17,31)(18,34)(20,36)(21,38)(22,32)(24,27)(25,37)(29,39)(30,40)(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=egh,e^2=hi,f^2=i,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, 104814);