#195414

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	#60951	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1152 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,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, 102610);