#219592

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	#3205	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,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,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, 126788);