#190225

order	512
nilpotency class	3
isoclinism	isoclinic class of #190224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1058 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fi,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, 97421);