#219556

order	512
nilpotency class	3
isoclinism	isoclinic class of #219552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

593 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^c=gh

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, 126752);