#199179

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1297 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)(11,16,22,32)(12,24)(13,18)(14,28)(15,19)(17,26)(20,36,25,40)(21,31,23,38)(27,34)(29,35)(30,39,33,37);(1,2,6,8,4,7,3,5)(9,11,15,22)(10,16)(12,25,27,20)(13,21,14,23)(17,33)(18,31)(19,32)(24,40)(26,39,29,37)(28,38)(30,35)(34,36);(2,7)(5,8)(9,12)(10,17)(11,20)(13,26)(14,29)(15,27)(16,30)(18,34)(19,35)(21,37)(22,25)(23,39)(24,28)(31,40)(32,33)(36,38);(9,13,15,14)(10,18,19,28)(11,21,22,23)(12,26,27,29)(16,31,32,38)(17,34,35,24)(20,37,25,39)(30,40,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106375);