#199396

order	512
nilpotency class	3
isoclinism	isoclinic class of #199230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1240 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)(3,4)(5,8)(6,7)(9,10,14,19,15,18,13,26)(11,16,22,31,21,32,28,30)(12,23,24,17,25,34,27,33)(20,35,38,39,36,40,37,29);(1,3,4,2,6,8,5,7)(9,11,13,22,15,21,14,28)(10,16,19,30,18,32,26,31)(12,20,27,38,25,36,24,37)(17,29,34,40,33,39,23,35);(9,12)(10,17)(11,20)(13,27)(14,24)(15,25)(16,29)(18,33)(19,34)(21,36)(22,38)(23,26)(28,37)(30,40)(31,35)(32,39);(1,4,6,5)(2,3,7,8)(10,18)(11,21)(17,33)(19,26)(20,36)(22,28)(23,34)(37,38)

Polycyclic presentation:

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

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, 106592);