#195054

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#34636	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102250);