#219598

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
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

591 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,14,21,28)(12,22)(13,20,30,26)(16,25)(17,32,34,29)(19,24)(23,37)(27,35)(31,40)(33,39)(36,38);(2,6)(3,8)(9,11,14,18,15,21,28,10)(12,22,38,19,25,16,24,36)(13,26,29,17,30,20,32,34)(23,31,40,37,27,33,39,35);(1,3,7,2,4,8,5,6)(9,12,14,24,15,25,28,38)(10,16,21,36,18,22,11,19)(13,27,29,40,30,23,32,39)(17,33,26,35,34,31,20,37);(2,6)(3,8)(9,13,28,32,15,30,14,29)(10,17,21,26,18,34,11,20)(12,23,38,40,25,27,24,39)(16,31,36,37,22,33,19,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126794);